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Today we're going to be looking at concepts relating to differentiation and we'll do a lot of other activities relating to that from the four past exam papers that we have. So to kick off, you need to be able to know how to differentiate because you need to know the basic rules of differentiation, which are straightforward. When you differentiate, you're going to multiply with what is in the power and subtract one from any value that is in the power. That is the differentiation. If you have a constant, for example, a constant is like a number one, two, three, four, hundred thousand, two thousand, five thousand. It's a constant. When you have only just a constant, a function with just a constant, y is equals to two, y is equals to hundred. That is just a function with a constant. Differentiating a constant is equals to zero, so the differentiation of a constant will just be zero. But if you have a constant times a variable, which then creates a function with a variable, differentiation works differently then because you're going to multiply with what is in the power of the function or the x or the variable, you will multiply that whatever is in the power with the constant. And you will have to go back and subtract one from the power. And that is the rule of differentiation. And if you're differentiating the sum of two functions x to x plus three y squared, those are two different functions, you differentiate them and add the differentiated functions. Now, when you work with differentiation as well, you must always go back to the powers, the rules of the power. Remember, any number without a power is to the power of one, so therefore it means when you are doing differentiation and of a function where it says f of x is equals to x, you must know that x is x to the power of one. Also, with the powers, when you do differentiation, you need to always remember that any value to the power of zero is equals to one. So when you do a differentiation and you have x to the power of one and you subtract one from the power which then the answer becomes x to the power of zero, you need to know that that function will just be the differentiated function will be equals to one because x to the power of zero is equals to one. The other thing that you need to always remember is in terms of the powers is the fraction. Remember, the root and the fraction. A root can be converted to a power of a fraction and a negative power as well can be converted to a fractional power, a fraction. So what I'm referring to, let me do this quick, because at some point, some way when we do the activities, you will experience the same thing. When I'm referring to if I have x to the power of negative one, you need to know that this you can write it as one over x is the same thing. In terms of the differentiation, if they give you a function that looks like this, you need to know that you can convert it into the power of a negative thing. The same similar if I have x to the power of negative two, this is the same. The negative only the negative can be converted to one over and the rest stays as x to the power of two. So the negative becomes one over and that is x to the power of one. And we know that when a value to the power of one is the same as that value. So those are the things that you need to be aware of. Similar to the root because the root is to the power of a fraction and you can move from a fraction to a root. Those things you will need to always constantly be aware of as you do differentiation. Okay, so those are the differentiations. Let's look at questions that we can do so that we can apply all these things that we reminded ourselves of. On the other thing about differentiation as well that you need to remember is when we are differentiating we are calculating the slope of a function. So meaning we calculate in the change in the values from the original values that they were to a different value. So you can calculate what we call the profit margin or the revenue margin or and so on. Okay, so the other thing you also need to remember is to maximize profit your differentiated profit margin function will be equals to zero and that will calculate whatever the value you are trying to maximize. So let's look at this question right. We all owned a Bureauceroll's tent. His profit selling Bureauceroll's is given by a quadratic function of y is equals to minus 40 squared x squared plus 90x minus 5323.5 where x represent the price of a Bureauceroll in rent. Vuo wants to maximize the profit to maximize the profit we need to make the differentiated profit equals to x. What should be the price per Vuo's B in order for Vuo to maximize his profit? So therefore it means we need to differentiate our y function and to differentiate remember we're going to look at all these values x is x to the power of one and this is a constant. So let's differentiate we multiply with what is in the power. So we have minus 40 times two because two is in the power and we're going to subtract one from the power that's the rule of differentiation and because this is a sum we're also going to treat the next one as a different function. So to differentiate this that will be 980 times one you don't have to do times one but I'm just going to demonstrate that x to the power of one minus one because I know that x is the same as to the power of one and I need to subtract one from the power minus differentiating a constant this is a constant because there's no x next to it or a variable next to it therefore it means it will be zero minus 40 times two is 80 it's minus 80 x to the power two minus one is to the power of one plus 980 times one is 980 x to the power of zero I don't have to include zero again this is the same as minus 80 x plus x to the power of one is one or x to the power zero is one so 980 times one is 908 and this is my differentiated function but that is not what we want we know that if we need to maximize profit the differentiated function of a profit should be equals to zero so therefore it means my differentiated function I should make it equals to zero minus 80 x plus 980 I'm sorry for the thinking of the thing I don't know why and how to stop that so we need to solve for x so let's move minus 80 to the other side it becomes 80 x is equals to 980 and because this is a function we can divide by 80 whatever I do on the left I must also do on the right therefore x is equals to 980 divide by 80 is equals to 12.25 and that's how differentiation works all you need to always remember is if they don't ask you just to just differentiate and they ask you to maximize profit when the question says maximize profit always know that to maximize profit you need to first differentiate and then make your differentiated question of function equals to zero and then solve for whatever the value that you need to be solving for are there any questions before we move to the next one let me stop right here actually and ask no questions okay if there are no questions then we can move to the next question on differentiation so question 20 and 21 are also differentiation questions so we need to answer question 20 and 21 based on the following information FM sell its product for 200 rent per unit the cost per unit is 80 rent plus x where x represents the number of unit sold per month so question 20 it says define the marginal profit function so yeah we need to define just the marginal profit function so you need to know some of the functions that's all some of them the formulas so in a way in general terms profit it's revenue minus cost right that's what we know so what will be our revenue and what is our cost in this instance so our cost we are told that it's 80 x so if our cost is 80 x it's minus times 80 x I can just listen to that and our revenue we are told that it is 200 per unit so our revenue will be 200 x so we need to solve sort out our profit 200 x minus 80 minus x and 200 then minus x is 199 x minus 8 and that is our profit function and we can also just the profit is a perfect function of x because we work in with x as a unit and therefore your revenue will be revenue of x minus the cost of x like that so therefore it means our formula here can also be changed to the profit function will be 199 x minus 8 which is option 4 let me know if you are lost or I'm talking drink so that I can explain to you can you please explain it again I'm a bit lost yeah so going back to the statement FM sells a product for 200 rent per unit right you can also even delete this so that we can start from the beginning so that's what they told us they sell so that is a sale sale equals to your revenue so 200 rent per unit the cost per unit per month is given by 80 plus x which then they gave us our cost of selling this product where x represent the number of units sold per month so you need to always remember that if you need to calculate profit profit it's given by your revenue or sales minus the cost revenue minus cost right the question in 20 says determine your profit function marginal profit function which is the same as profit function because at the moment we haven't calculated the marginal profit function as yet because the marginal profit function is a differentiated function so the sale we know that it is 200 per unit so it means it's 200 times x amount will give us a revenue so if I have to sell 10 this is how much revenue I would have brought to 10 times 200 would have bought the revenue of 2000 but because I don't know how many products I would have sold but I know that I will be selling them 200 of the x amount of the units that I don't have I don't know this will be my revenue the cost they told you what the cost is because remember the cost pay unit is the the cost is a and yeah because they say also cost pay unit or maybe because I should have thought about it this way I didn't think about it in that way because yeah oh yeah I think your question brings a good thing into mind because the question here is profit margin and let's sort out that part first so this is the cost pay unit I only took it as the cost pay unit as that but it needs to be minus the cost pay unit will be your 80 plus x which is the cost pay unit times how many x amount of units that you will be selling so it's 80 plus x of the cost of those units that you are selling because we don't know what how many plus how many so I think I the first time I did this I did it all wrong I got too excited to see the answer one day minus x times becomes minus 80 x and minus times minus is minus which is plus x squared plus x squared now the answer here will be 200 minus 80 which is you can see that it's totally different to what I had previously right plus x x squared that is the profit function and now to answer the question that they are asking us which is marginal profit right that's what they are asking we need to calculate the marginal profit so to calculate the marginal profit we need to differentiate because marginal profit is your differentiated profit marginal profit is your differentiated profit function so to calculate the marginal profit we apply the same rule we differentiate this multiply by what is at the power which is 120 times 1 x 1 minus 1 plus multiply with its with what is in the power it's 2 times x 2 minus 1 therefore our marginal profit will be 120 x to the power of 0 plus 2x to the power of 1 which is the same as 120 plus 2x which is number one I apologize I totally apologize for misleading you the first time because I didn't take that into consideration okay so this is how you will find the marginal profit because marginal profits are your differentiated profit function so question 20 it's option one 21 it says what is the marginal profit if the production line is 20 units so they're asking us what will be the marginal profit the differentiated profit if we sell 20 units if we sell additional 20 units therefore 120 plus 2 times 20 because we need to substitute the 20 into the x value and that will give you 120 plus 40 which is 160 my calculating right is 40 40 plus 120 is 160 yes calculated right you guys is it a long Saturday or what much using the wrong I selected the right formula but I'm using the wrong the wrong sign should be minus all over not a plus I wish I can restart the session so that it makes sense because then starting on the wrong footing it's not a good thing at all because the answer is 120 minus 2x so it's 140 120 minus 40 which is equals to 80 which is option oh gosh any of this but that's how you will do it before initiation find more questions hopefully this one will be able to do them correctly one time see if we can find yeah a company total profits from producing and selling x units of its product is given by the profit function of minus 0 comma 0 2 x to the power of 2 plus 300 x minus 2000 200 000 how many units of this product must the company produce in order to maximize profit so this is for you to do maximize profit we need to first differentiate the profit function so you first need to differentiate this profit function and once you have differentiated the profit function therefore you need to make sure that the profit function is equals to 2 and calculate and find x that is for you to remember to differentiate to multiply with what is in the power and subtract one from the power and also differentiating your constant it's 0 can we do it together yes okay let's differentiate we're going to multiply with what is in the power so minus 0.02 times 2 times x to the power 2 minus 1 plus 300 times 1 x 1 minus 1 because x to the power of 1 is missing then it's the same as x and minus 0 because 200 000 is a constant so it will be equals to 1 and this will be minus 0 comma 0 4 why this thing is freaking up you know x to the power of 1 I don't have to write it plus 300 times x to the power of 0 x to the power of 0 is 1 is the same as 300 and now I know that because they talk about maximizing profits so I need to make the differentiated profit function equals to 0 so therefore 0 is equals to minus 0.04x plus 300 which is equals to taking minus to the other side it becomes 0 comma 0 4x it's equals to 300 divide both side by 0 comma 0 4 0 comma 0 4 0 comma 4 0 comma 4 cancels out therefore your x will be equals to 300 divide by 0 comma 0 4 it's equals to 7500 happy yes okay so let's look at another question question 19 and 20 are based on this same information here they have given you the average cost function for the product are given by this where x represent the number of units sold how many products units were sold oh sorry how many production units will result in a minimum average cost per unit they want to to show them how many how many how many um production units will result in a minimum average cost per unit so we can differentiate the product are they really can we yes we can differentiate the the cost function and then make the differentiated function equals to 0 and then calculate what will be the average cost now you have 1 over 4 x plus 4 plus 100 over x so over x is the same as x to the power of negative one that's given 2 over x let's assume that I have that this is the same as 2 times 1 over x because if I multiply 2 with 1 because 2 is the same as 2 over 1 multiply what is the numerator with numerator so it will be 2 times 1 is 2 1 times x is x so the wait means I can also rewrite this 100 over x this way and I can also rewrite 2 times x 2 times 1 over x is the same as 2 x to the power of negative 1 because if 1 over x is the same as x to the power of negative 1 why not so let's do the same with 100 so 100 over x will be 100 times 1 over x which is the same as 100 x to the power of negative 1 right so we can rewrite this whole cost function as 1 over 4 x plus 4 plus 100 x to the power of negative 1 so we can write it as over 4 x plus 4 plus 100 x to the power of negative 1 so now yes differentiate this differentiating this function will be x is the same as x to the power of 1 so therefore it will be 1 over 4 times 1 x to the power 1 minus 1 plus 0 because 4 is a constant I'm not going to put oh I can put the plus 100 times minus 1 x to the power of minus 1 minus minus 1 because I need to subtract 1 from the power I don't know why my screen keeps on flicking and I've tried to minimize maximize okay so let's differentiate that will be 1 over 4 x to the power of 0 will just be 1 and minus 1 times 100 and 0 we can just ignore that minus 1 times 100 is minus 100 and x to the power of minus 2 minus 2 to the power of minus 2 and x to the power of minus 2 this whole thing we can rewrite it as we can re-change this minus but we list that it's known so which is the same as 1 over 4 minus 100 over x squared if we need to maximize or minimize we make this equals to 0 and if we do that then we can take x over what do you call this now minus 100 over x to the other side it becomes positive it will be 100 over x squared you must remember that this we solve them as the same way as functions right so equals to 1 over 4 and we treat this as a function we need to only have x over 2 on this side so it is a it is multiplying or it is dividing so it means we need to multiply by x over 2 on the other side and we also need to multiply by x over 2 on the other so this side x over 2 will cancel out it will be left with 100 in this side you will have 1 over 4 x squared I know that it's something that keeps on changing around a lot but to get rid of 1 over 4 we multiply we do the we multiply by the inverse of the function right so it means we multiply this side by 4 over 1 therefore it cancels out and also this side we multiply by 4 4 over 1 multiply by 4 over 1 then this will cancel out then you are left with 400 is equals to x squared and to get rid of the 2 we can take the square root on both sides and 2 and the square root will cancel out and what is the square root of 400 it's equals to 20 x is equals to 20 so that's 20 units the long calculation run are we good? yes now based on that information now you need to answer the following questions it says choose the statement that gives a practical interpretation of if we find the cost unit to be 14.25 the average cost unit to be 14.25 what is our interpretation? so now you must also think of it in this way at this point because this is the average cost unit not the marginal cost unit so therefore it means you cannot use the marginal cost unit for it that's one so you will have to interpret it directly based on the average cost unit so which one will that be is it number two number three number four is it number two? it will be number two because it is a cost unit interpretation and it is the average cost unit interpretation so this is not the total cost or it's not the fixed cost it is the average cost so the average cost unit of producing 25 units because if you come to this function and put the 25 if you calculate this and you get 14.25 you just need to make sure that you interpret it based on that so the average cost unit of producing 25 units will be 14.25 that is the average cost which is option number two okay if they would have given you a marginal cost unit like that then you will do it that way as well okay so let's go to the next question paper see if there are more questions as well the other thing is because they don't tell you that differentiate you need to be able to know how to identify the question in order for you to know that this is the question to differentiate but on this one they did give you a statement to say differentiate this function which makes it easy because now you know that you need to apply differentiation so differentiate that statement when I go get water that is your exercise I'm going to give you some few minutes do it on your own and I will do it with you now we have the answer oh because it's got fractions let me know if you need help please have help with the fraction part of the sub epsilon hundred percent show on that okay let's do it together let's differentiate okay it will be six times three x three minus one right minus 12 times minus two x to the power minus two minus one plus four times three over two x three over two minus one minus zero and that's us straightforward six times three is 18 x to the power of two three minus one is two minus times minus is positive 24 because 12 times two is 24 x to the power minus three let's calculate the fraction so you can simplify two can go into four it goes into four how many times two times right so you can say two goes one time and it goes two times two times three it's plus six x and then you are left with the fraction at the top to calculate so the fraction at the top is a subtraction so we need to find the common denominator so it's three over two minus one so the common denominator is two you say two goes how many times into two it goes one time one times three is three one goes how many times into two it goes two times two times one it's minus two then the answer here will be minus three minus sorry three minus two is one over two so therefore x to the power one over two will be the answer um Elizabeth do you not then have to differentiate it further no you only do it one time one time okay and then that's it so now we you can rearrange to look at which one so let's see because you can see that this one has 36 this one has a four and this one has a six so let's see 18 six and 24 but the powers are different and this one has 18 to the power of two plus six to the power of a half plus 24 to the power of a negative three and then that's it you only do the differentiation once one time once off you don't go and differentiate again after you get this answers one time only so the answer is option one let's see if there is another question we might even finish earlier than two o'clock because we are left with one question and then we can just question an answer for any other question that you might have so the total cost of a manufacturer x watch is given by a function and this is the cost function the total cost function of a thousand plus hundred x minus x squared over four the marginal cost to the manufacturer the 35th watch is so they need you to calculate the marginal cost of the 35th watch not the not maximize profit right so you just need to make sure that you differentiate this because remember marginal means differentiated function so we need to differentiate the cost function and you also need to remember that x squared over four is the same as one over four times x squared right anyway yeah I want in the same thing so differentiate this cost function are we finished so this function you can just rewrite it as a thousand plus hundred x minus one over four are you winning are we winning do you need help do you just end up with minus a quarter x to the power of two plus a hundred you don't just end up with the minus a content because remember next to x there is much to the power of x is to the power of two right what did you do with that too it needs to multiply the quarter right must multiply with what is in the power and subtract one oh yes don't forget that very important so you get minus or half x do you then substitute the 31 with the x yes okay are we winning are we there yet so let's see so let's differentiate differentiating a constant that will be zero plus yeah is to the power of one so that it means it's hundred times one which is hundred x to the power one minus one minus one over four times two x to the power of two minus one x to the power of zero is one so our answer will be hundred times one is hundred two goes one time into two and it goes two times into four so therefore yeah we'll have minus half x to the power of two minus one will be equals to x to the power of one which is x in order to find the marginal profit after producing the first watch we will get hundred minus half times 31 which then it is equals to 84.5 which you can say it's equals to 84. Happy are we good are we good oh good oh good days nice let's look at the last text and paper and then we can find more I will I do have more other activities or exercises that we can do if some of the questions in this are included in that exercise that I have selected then it's still fine we can skip them so we're looking for differentiation functions see my Siflac on this one there are no differentiation here we do okay so this one I do have it on there um we'll just exclude it from the activity that we do so sells bottled water at a special price the bottled water cost the store three rent each with a fixed cost of 280 per week for marketing the weekly cost is linear function of a unit price x given by this so this is our cost function because they gave us our cost function so yeah they didn't just say this is just the cost of x per unit so we know that this is our cost function the revenue function as well is given by from selling the water bottles is given by the quadratic function which is that revenue function and they say if your profit function is given by your profit is equals to your revenue minus cost then find the marginal profit function so what I'm gonna do I'm gonna copy this and because I'm tired of the the PDF flipping around as often as possible let me just copy this and paste it onto a power point slide and then we will answer this from from there and then I can just share the power point slide because I know that they it's not going to flick around it will be say a new slide and share there and I'm gonna check if there are more questions on this yes we do I'm also going to copy this shelf I'm gonna also paste it on because it's not nice to look at these things flicking all the time yeah because of sharing here that our points tonight the state of the PDF document and we can do the activities from them won't work with me huh if we can do it this way okay are you able to see this glance okay so this is the same question let me make you on now it's easy all right in here and there is no disturbance okay so now the first thing we need to do is create the profit function based on the information given so we need to substitute these equations into this function so that we can then calculate our marginal profit so the first is to substitute your revenue function which is minus 20x squared plus 500x minus and I'm going to put this in a bracket because it is the entire function and we know that the equation has a negative so 280 minus 3x simplify the equation how do we simplify and this will be minus 20x squared plus 500x minus times 280 it's minus 280 minus times minus positive 3x and like them together right let's do that minus 20x squared 500x and 3x we add them together it will be plus 503x minus 208 now we do have now our profit function but what we want is the marginal profit function which is our differentiated function so I'm going to give you a few minutes to do your differentiated profit function based on our last function so this is our profit function differentiate that and let me know if you need help should be quick are you done yes is the answer what is your differentiated profit marginal profit minus 40x plus 503 yeah so it will be two times minus 20 which is minus 40x 2 minus 1 will be 1 so it's the same as x and differentiating 503x is the same as plus 503 differentiating minus 280 it's equals to zero so we don't have to write anything so that is your marginal profit and the answer is option two there we go let's move to the next question find the derivative finding a derivative is the same as they are saying differentiate all right so we just need to differentiate this remember you have 3 over x is the same as 3 times 1 over x which is the same as 3 times x to the power of negative 1 so we can rewrite this function our f of x oh sorry not our f of x our function is equals 2 because we just rewrite in the whole thing 3x plus 5x minus 3x to the power of negative 1 differentiate now you can apply the derivative so it can be quick let me know if you need help and let me know if you are done I'm done yeah is the up is it option number one minus 3 plus 15x to the power of 2 plus 3x to the power of 2 I don't know because I haven't worked it out so let's see let's work it out so differentiating 2 will be equals to 0 differentiating minus 3x it's minus 3 because x 1 minus 1 is 0 which it's multiplied by 1 plus 5 times 3 and 15 x to the power 3 minus 1 2 minus 1 times minus 3 plus 3 plus 3 x to the power minus 1 minus 1 minus 2 minus 2 so therefore the answer will be minus 3 plus 15x squared plus 3x minus 2 which one it is number two no number three number three because number three is oh sorry it's number four it's number one no it's not number one number four it is number four yes minus 3 plus 15 squared plus 3x minus 2 yes number four is the correct one other questions the function of the total profit in thousands of rents to sell x item is given by this profit function what is the marginal profit differentiate this function that's your question all you just need to do is find the differentiated function and then once you have the answer you can rearrange the date the the information okay it was easy right right differentiating your constant is zero yes differentiating the next one so that I can remind those who have who forget because I think because we've been taking shortcuts let's do it step by step this time we multiply 100 by 1 and subtract 1 from the power minus 5 times 2 x 2 minus 1 therefore this will be 100 times 1 is 100 x to the power of 0 minus 5 times 2 is minus 10 x to the power of 1 which is the same as 100 times 1 which is 100 minus 10 x we can rearrange this minus 10 x plus 100 is our function which is option number moving on to more complex differentiation function and you look at this and you'll be like scratching your head oh no where do I start where do I even begin easy right you know the powers you know the roots everything you know about the basic things apply this let's rewrite the whole but we will rewrite it in different steps so that you can we can know what we do so let's start with this part first I'm going to use the bottom and I'm going to scratch it out or let's not stretch it as I'll use the top part here so we have two times the quadro root of g to the power of three now the first thing we need to do is get rid of this quadro the the root side right let's do the root first so we know what do we know about the root because in front there is a two we say it is to the power over half we can do the same the with the four it will be the same as to the power of one over four right so let's do that we change this to two times g to the power of three to the power of one over four so now we have now we have fractions one over four times now we have one over four times three over one because that's what and the power is because the rule of the power says if it's any power or any value to the power multiplied by the power we can just multiply the powers so we're going to multiply the powers three times one is three one times four is four so this we can rewrite it as two g to the power of three over four right so let's write it back onto our table here we can rewrite it here or i've got another problem with the ink is two times g to the power of three over four now let's solve the next part which is this one so also say it is one over if not the minus because the minus is there so one over four g to the power of three right we can split this into multiple parts so because it's we can split one four g as one over four times one over g three right because one times one is one four times g three is four g to the power of three right so we can split it in that way so once we have split it in this manner then the next logical thing is to convert because numbers are fine so we don't have to worry about the number so we can come back and write the number here and only have g uh one over g to worry about so we already sorted that one part so we have one over g to the power of three to sort out we know that this we can rewrite it because we know that one over x we can write it as x to the power of negative value that's what we know and we also know that x to the power of negative two come on x to the power of negative two we can write it as one over x to the power of two which is similar to what I have there right so we can move from one over g three to this it can be written as g to the power of negative three the same thing right this and that we rewrote it the same one so the next part is g to the power of negative three plus ten to the power four don't be fooled by ten to the power four is still a constant so now let's differentiate because I've rewritten the whole equation in a way that it enables enables us to do differentiation so let's differentiate not not g but f we need to differentiate f of g which is easy multiply with what is in the power so two times three over four now I've got another issue with my ink oh gosh g to the power three over four minus one minus one over four times minus three g to the power minus three minus one plus zero because this is constant it's because is it two goes two times into four so therefore I have three over two as the answer and we need to solve the fraction so three over four minus one my common denominator is four four goes one time one times three is three one goes four times four times it's minus four therefore the answer here will be minus one one over one over four so my g will be minus one over four as you can see already they should be jumping up and down because we have an answer some way but you can see that really we're not going to get that answer it's not right okay so moving on minus times minus is positive and three times one is three over four and g minus three minus one is minus four now don't get too excited because if you look at this we know we clearly can see that this is not the right answer so three is not it's not going to be our answer so it means we need to take back everything we just did back to the root in order for us to get the answer correctly here okay so if we need to take it back we also need to do it step by step so let's let's do it step by step let's start with g to the power of negative g to the power of negative one over four you remember what that is right it is yes it's the safety so we can just do that three over two times the quadruple root of g that sorts one thing out right it sorts one thing out then you can be too excited about that okay and also if I look at all the answers it doesn't seem like it will be the right the final answer but we'll get back to it just now and what is g to the power of negative four we can write it as one over g to the power of four right so plus three over four g one over g to the power of that now because all of these things are multiplying you can see right they are multiplying to one another so like the other thing I've I made this is wrong because this is wrong I'm gonna tell you why it's wrong right now we have the quadruple root of g which is g to the power of one over four but this is minus so it cannot be that what we had so we need to say because it's g minus one over four regardless of whether it's one over four we need to say one over g of one over four and only convert the g over this one at the bottom to one over the quadruple root of g so the answer should be like that so this is multiplied by one over g something like that now we can simplify it further as well so three times one it's three over two times two times four g plus three times one is three over four times three g to the power of four which one is the correct answer let's see it's option number one three two four three three three in state of four it's times four four to the power g and that's how you do differentiation which is option one it's a lot right I hope they don't give you questions like this in the exam and they give you questions like this right so this looks almost similar to the previous one that we did oh you do wanna maybe I need to wait here so that you can take a screenshot to take a picture or something uh and you can always come back to the recording yeah to look at all this okay so exercise three a certain a certain production facility the cost function is two x plus five and the revenue function is eight x plus x squared and they want to to find the profit function so you just need to remember that the profit function it's given by your revenue function minus your cost function so just substitute the values and remember there is a negative sign so it means the cost function you will put it into brackets to accommodate that negative and simplify and see if you get the profit function and if you are done let me know you can still have 20 minutes let's see in it still be quick and easy and you can also rearrange the values just solve the functions let me know when you're done oh you guys are posting on the on the chat sorry I've been not been looking at the chat um are we done do we have an answer okay no question no response yeah x minus x squared minus into bracket two x plus five so let's simplify the function it's negative two x negative two x eight minus two you're eight x minus two x it's plus five it's plus five no but do you need to multiply by the negative um um Joan you're you're you are echoing us I'm not sure why it's as if you have another listening device somewhere close by okay so it is like times together why why do I have two x it should be just x not minus two x it just it's eight x minus x squared minus two x times minus then that minus times positive would be minus five and like times together so minus x squared eight x minus two x is positive six x minus five the answer is option so the next question what will be the amount that must be produced to maximize profit remember maximizing profit it means you need to find the marginal profit and make marginal profit equals to zero and therefore find your x vehicle so it means take your profit function and differentiate it so differentiate your profit function you can do that together so to differentiate this we multiply with what is in the power minus two x two subtract one plus six x one minus one minus zero which then it is equals two marginal profit will be minus two x plus six and because they say maximizing profit therefore we make the marginal profit equals to zero and solve for x take two to the other side two x to the other side it becomes two x is equals to six divide the side by two therefore we divide on the other side by two and that will be equals to three and remember it is to the thousand multiply that by thousand and the answer will be three thousand units amount of units the answer will be option four okay are there any questions anything that you want me to verify or you still unsure of the next last couple of questions should be able to help you clarify some of these things because then there is nothing more I can offer you except give you answers which is not even going to be helpful at all so calculate the marginal profit cost oh marginal cost function means differentiate this cost function and you can when you are done you can tell me which option is it one two and three and I can show you how to answer the question as well okay okay maybe there is no no this is the last maybe option four option one maybe option one I like the the fact that you're saying maybe it's like if you're not sure of your answers guys okay let's see let's see let's see remember it's always about it's about multiplying with what you have so one over ten times two remember two is two over one let's put it this way maybe if I explain things in this manner some people might understand them better times x to the power two minus one plus five times one x to the power one minus one and a thousand it's always zero regardless of how big the number is it would be zero so two times two times one all we can say two goes one time into two and it goes five times into ten and therefore one times one is one five times one is five x to the power two times one oh two minus one is one minus two plus five times one is five five times one is five and the answer is option one then the next one based on the same information if we have a thousand additional units to produce interpret all this like I'm saying it you can hear in my explanation thousand a not thousand then it's equals it's thousand additional more adding more adding additional more items will reduce or increase so let's find out so if we add c of a thousand into this function five times a thousand one over five times a thousand plus five what do we get plus five is two hundred and five so one additional unit will be equal to that so if we add a thousand additional units one additional unit or if we add one additional thousand kilogram unit of whatever the thing we are producing of steel it will yield the cost of two hundred and five that's that's how you interpret this so you can you cannot say it directly so only directly is when you are using cost function not the marginal cost so when you're using your cost function you can say things directly to say if I substitute the thousand in the cost function I can say when the cost of producing a thousand kilogram will give two hundred and five but because I'm not interpreting the cost function I'm interpreting the marginal cost function which is the slope and the slope we always say one additional unit either you are increasing or decreasing because it's the slope if it's positive it's increasing if it's negative it's decreasing so one additional unit of producing a thousand a thousand more kilograms will yield or will produce two hundred and five so this will be how you will interpret it always remember that to always use the weight like this it will increase by it will yield it will produce one additional unit it's the slope slope know how to interpret the slope of the equation okay if given the marginal profit so this one they have calculated the marginal profit therefore it means they're giving you the differentiated function if the marginal profit in thousand or friends to produce a toy motorcycle is given by this where x is the number of toy motorcycle produce find the marginal profit for producing 20 additional units and interpret so it means at the end we need to go and interpret the answer so let's see how do we interpret and how do we calculate that so it's easy because we told we need to calculate for 20 more units minus 20 divided by 5 because we just substitute the way we see x we put 20 so the answer would be 30 minus 4 is 26 how do we interpret when 20 motorcycles are produced there will be an increase because my it will increase in production will yield an increase of 26 it shouldn't be an increase it should be a decrease actually because of the negative but anyway it will yield an increase of 26 run profit but our slope function here is negative I don't know why it would have been an increase anyway your last question differentiate this function minus 7 plus 5x minus 5 over 2x quit minus 3x to the power 4 and this should take us to the end of the session so let me know which option you think since it's our vocabulary it might be one or two or three or four I have the answer Elizabeth let's see what do we think it is no response I'm still busy I think the answer is number three let's see if it's number three differentiating a constant is 0 5 times 1 it's 5 x 1 minus 1 is 0 3 times 5 minus 10 times it's 10 times 10 over 2 x 2 minus 1 is 1 3 times 4 it's minus 4 minus 1 is 3 let's rewrite it properly this will be 5 minus 5 because it's 5 times into 10 to the power of 1 so it's x minus 12 x 2 so it is option three yeah option three it is I had one last exercise we probably we can do that you're not in a I'm also not in a hurry today so we can I can show you the last exercise as well to do but it almost look now this is the same exercise we did sorry it is that exercise this exercise yes it's one and the same thing so I'm not going to repeat it so anyway then that concludes today's session because there are no more other exercises I have for you are there any questions I see that someone has posted there can I have access to the the recordings I'm going to show you now I'm going to share with you as well in the link in the chat let's see if I can find the link okay yes I do have the link I'm going to share the link in the chat please uh go watch other recordings from there and I'm going to also share or not share but share my screen I'm just gonna stop sharing on this one so that I can share what I want to share with you just now to answer that question so that then even in the future it is on the recording so that nobody can unless if people don't watch their recordings okay so please make sure that you also complete the register I'm gonna post the link as well in the chat just now we need to complete the register the link to the register and I explain what I'm just sharing with you just now in the link to the register is in the chat as well so please make sure that you complete that otherwise you can put your student number on the chat and you don't remember if you put your student number there you are consenting to publicly share your student number on the public platform like on the chat okay so what I'm sharing with you is where you can find the recordings not only for today's session but for any other if you didn't know UNISA Western Cape has on my UNISA or my module platform where we save all the recordings for all the sessions that we offer to students so for tutorial classes if you have any of the tutorial classes or modules that offer tutorials you can come here and look at their recordings at the writing center for English mostly you can check if any of your modules are on here and go and watch the recordings for numeracy center where we are at there I've been with UNISA students since the beginning of the year but for second semester we do have recordings so I'm just gonna show you if you click on that link it will take you to all the weekly recordings that we had for all the Monday sessions that we had up until the 19th of September so all the recordings are are here for the Akali sessions or numeracy center sessions and classes that we had it takes time to reflect but there are the you will find all the recordings there the topics are generic so it's not QMI specific but you will see that they might be helpful if you need the notes linking to those recordings they are always under this open class folder there are notes in there if you click on it it will open another tab where you can go and find all the notes there will be notes for first semester session classes that we offered which some of them might also cover like topics like differentiation I can't even remember now all of them but the notes are in here and then for today's session I just want to go to today's session there is an exam prep link also which is the QMI examination preparation link here at the bottom if you click on it it should have all the recordings for the past two sessions that we have including today's session it will be loaded on here probably you can check next week okay so they are this here are the recordings for the last two sessions that we had also the I think what you call this thing now the past exam papers that we used in the session I think I've uploaded them on here so let's see yes there they are so the four past exam papers that we just shared now with you and went through the the questions they are there to help you prepare for the exam what else do I need to share with you nothing else there's nothing more other than what I've just shared but for example if you if you need some of the recordings for the semester one that we did you want to go back and look at them you can check my youtube channel I think it's called it's my name my full name you can just google me I think maybe now I'm even famous on youtube I'm joking I'm not but you can go there there are some of the recordings that we do so I did you can go and look at them you can search for QMI and it should bring back it's it will be the same recordings that are uploaded on on Unisa's platform it's not something you so yeah you can search there you will find more of the recordings we we do have last year's recordings as well where we did the exam preparations and all that you can go through that as well but everything it's on it's it's available for everyone so free of charge of cost because all Unisa work that we do we do it to support you guys so use that then you do have my email address if maybe you don't have it it's eboyematunisa.ac.za and if you need to have a consultation with me we do have a consultation platform where you can also ask for a one-on-one consultation but a my business schedule sometimes doesn't allow me to but yeah so other than that if you have any questions or comments you can send me an email or copy ctntat at unisa.ac.za and we will shortly answer or respond to your questions yeah that's it there's nothing more I can offer you I will see you next week when we do our last session in terms of exam preparation we will be looking at financial meds so bring your financial calculators if you haven't bought a financial calculator now is that chance that you go and buy that financial calculator so that you allow yourself time to practice with it and I think the two hour session will allow us to do lots of activities and lots of exercises but because there are so many financial meds questions yeah if you have a financial calculator it will save you a whole lot of time in the exam as well when you answer those questions so I will see you next week and our final final session of the year all the best to those who are writing their exam this have already started or they are starting this next week all the best until next time next same place same time thank you bye bye guys see you next