 In this video we want to develop some more properties of the derivative. For example if f is differentiable that is to say that it has a derivative f prime of x exists at the point and c is any constant number then the derivative so d dx of c times f of x is just c times the derivative of f of x. So in other words if you take you know a number like two times f of x and you're taking the derivative of this so just to be two times f prime of x. So that is you can factor the constant multiple out of the derivative process and this is a natural consequence of limit laws. So if we have our constant c and we take the function c times f and we want to calculate its derivative then by the definition of the derivative this is the limit of the difference quotient where we're going to take cf at the point x plus h and subtract from it cf at the point x and then we divide everything by h where we have to cancel the h and the denominator if we can right but what is the function cf of a number even mean like if you take you know cf and evaluate it at two what this means is just you're going to take c times f of two you just take the function evaluate the number then times it you know as an afterthought by this constant and so when you take cf of x plus h this just means c times f of x plus h and same thing here cf of x just means c times f of x now this is a constant multiple which could be factored out right we can factor it out of the numerator so we get c times f of x plus h minus f of x this all sits above h as h is going to zero but as now this is a constant multiple of the limit it could be taken out of the limit process so it sits out right here but then this right here is just the derivative of f evaluated x and since the function is differentiable this limit exists and is equal to f prime of x so taking the derivative with the constant multiple means you can take the constant multiple out when you combine that with the power rule that then allows us to calculate some more types of functions like take y y equals eight times x to the fourth well by this constant multiple rule if we take y prime that is we want to take the derivative eight times x to the fourth by the constant multiple rule we can take the eight out and so we have to take the derivative of x to the fourth which by the power rule that's going to be four x cubed you bring the coefficient out in front that is the exponent becomes the coefficient out in front and then you reduce the coefficient or the exponent by one then you're going to take eight times four which gives you 32 x cubed that would be the derivative of this function how about this one right here if we want to take the derivative of y which equals negative three quarters times x to the 12th well when we calculate dy dx here we're going to be taking the derivative of negative three fourths x to the 12th for which we can pull that coefficient out we can negative three fourths and then we take the derivative of x to the 12th by the power rule that gives us 12 x to the 11th for which then we'll multiply the coefficient that was already there with the new coefficient we got from the power rule four does go into 12 three times so we get negative three times positive three we end up with a negative nine x to the 11th as the derivative of this function looking at the next one if we take the derivative here of negative eight t which we're going to be taking the derivative with respect to t right here again take out the negative eight then we have to take the derivative of t which we saw by the power rule if you take the derivative of just a single variable you're just going to get back a one and so you end up with negative eight as the derivative it's a constant here and so this is something you see very quickly that since the cost of multiples just you know just factored out of the process we kind of can start skipping steps I mean the whole point is this process is speed up the calculation make it more efficient so when you look at something like y equals 10 times p to the three halves if we take the derivative with respect to p dy dp we're going to end up with 10 times right then we take the derivative p to the negative three halves power which by the power rule gives us three halves p to the one half power where did one half come from well one half is three halves minus one and so since you're just going to factor the 10 I'm just going to leave it there I don't need to do it in two steps I'm just going to do it all at once and as two goes into 10 five times and five times three is 15 we see the derivatives as 15 times p to the one half power so combining these these derivative rules we can speed up this process very very quickly how about this one right here let's take the derivative of y equals six over x which to maybe make it easier it's better to see this as a power function so y equals six to the x times x to the negative one so then by the power rule with this constant multiple rule y prime we equal six times negative one times x to negative two remember we subtracted one from the exponent and so rewriting it back as a reciprocal function we end up with negative six over x squared as the derivative this constant multiple rule works out very nicely now let's combine it with another derivative technique this is sometimes called the sum rule which I don't mean like in Charlotte's web this is some pig I mean this is the sum rule right and which tells us that if we take if we have two differentiable functions f and g they're both differentiable at a point then the derivative of f plus g of x will equal the derivative f plus the derivative of g so if you have two functions f you know you have f of x plus g of x and you want to take the derivative this will equal f prime of x plus g prime of x you can separate two functions connected by a sum and take their derivative separately all right it's also true you can do this for differences if you have f of x minus g of x you take its derivative this will be f prime of x minus g prime of x the proofs of the sum rule the difference rule are basically the same let's look at the argument behind the sum rule just for a moment because again f minus g will be handled very similar so if we have f plus g prime of x well by the definition of the derivative that is the derivative is the limit of a difference quotient this would look like f plus g evaluated at x plus h now this looks like we're supposed to foil but this notation might be a little bit misleading here we're not multiplying here the name of the function is f plus g and the name of the operands we're putting into the function is x plus h and then we have f plus g evaluated at x all over h well what does it mean to have a function f plus g evaluated at number x it just means evaluate the evaluate the function at the number f of x evaluate also the number at the function g and then add together whatever those outputs are so when you add together two functions you're just adding together the output and so that tells us that f plus g at x is f of x plus g of x this also tells us that f plus g at x plus h is f of x plus h plus g at x plus h which when you look at that it's like it kind of looks like you distributed x plus h here so i i guess it does kind of look like it's a foil i mean it's not but it has the same notational similarities now so we have an f of x plus h plus a g of x plus h minus a minus a f of x and a g of x which notice this negative sign was distributed on to the f of x and to the g of x so if we do distribute that through we're going to end up with an f of x plus h we get a g of x plus h we're going to get a minus f of x and we're going to get a minus g of x this all sits above h as h is going to zero for which i'm then going to regroup things let's put the f's together and let's put the g's together for which you then see this f of x plus h minus f of x and then you see this g of x plus h minus g of x in the numerator of our difference quotient but if you had something like one plus three and this sits over five you could break this up into one fifth and three fifths that is i'm just going to break up this fraction into two fractions that is to say this then looks like the limit of f of x plus h minus f of x all over h and then we're going to add to that g of x plus h minus g of x all over h and so these are all sitting inside of this limit so we're going we can broke we broke up our original fraction into two fractions which these two fractions are themselves difference quotients well if you take the limit of a sum right so we have a we have a limit here and a sum of two things by limit properties you can break this into a sum of limits right here so we get the limit as h approaches zero of f of x plus h minus f of x over h and we also get plus the limit as h approaches zero of g of x plus h minus g of x all over h for which when you look at this by itself hey this looks like the the derivative this is just the definite derivative of f and since f is differentiable this limit exists and is equal to f prime of x likewise this here is just the limit of the derivative of this limit here is the derivative of g so it's just g prime of x and so we see here that the derivative of the sum is the sum of the derivatives the essentially the identical argument will work with negative signs here you just have to change all the negatives so this becomes negative this becomes negative this becomes negative this is a negative this is a negative these go through all of this becomes a negative this becomes a negative this becomes a negative there's the proof for f minus g as well and this follows from limit properties that we saw previously in chapter two of our lecture series now i do have to caution you that products and quotients will not be so simple we will actually approach those in the next lecture i also want to point out that if we take the properties we've learned so far the power rule the cost of multiple rule sums and difference rules we actually have the technique to compute the derivative of any polynomial function let me show you how this would work so here's a perfectly good polynomial function y equals 6x cube plus 15x squared if we take the derivative here y prime well we're going to take the derivative of this function 6x cube plus 15x squared but we have the sum of two terms 6x cube plus 15x squared which by the sum rule which we learned previously thanks to charlotte we get 6x cube prime plus 15x squared prime we can take the sum of these things separately and if we have more than two terms like we'll see in a second you can just take each the product that is take the derivative of each term in the sum separately then we can take out constant multiples by the constant multiple rule 6x cube prime plus 15x squared prime and then taking by the power rule we're going to get 6 times 3x squared its derivative and the derivative of x squared will be a 2x and then multiplying things together you get 18x squared plus 30x which is the derivative of this polynomial much simpler than we'd have to do it alternatively of course i think that a lot of this process can be simplified dramatically eventually we will reach a point where we'll say things like y prime is equal to 6 times 3x squared plus 30x right you'll just kind of multiply the things together you might be like oh 15 times 2x but you want and then the next steps multiply but many of you will be able to do this in one line y prime is equal to okay 6 times 3 is 18 times x square we lower the power and then you get 15 times 2 which is 30 and then you lower the power you get x many of us will be able to very quickly get to the point where the derivative of a polynomial will be a one liner this will be basically like a multiple choice question on an exam when this you know comparably on exam 2 when you're trying to calculate the derivative of such a polynomial from the definition would be probably be one of the hardest problems on that test you can see the huge advantage we could have from doing the power rule here with these other techniques let's take the function p of t equals 12 t to the fourth minus 6 times the square root of t plus 4 over 5 over t excuse me it's not a polynomial but we can make it into something similar to a polynomial that is we can make it into a combination a linear combination so to speak of these power functions and so we get 12 t to the fourth minus 6 I'm going to write that as t to the one half power and then for the next one we're going to get 5 times t to the negative one so then when you take the derivative the derivative of p of t that is p prime of t which by the power we're going to get 12 times 4 which is 48 lower the power by 1 we get t cubed the next one we're going to get negative 6 times one half which is negative 3 times that by t to the negative one half I lower the power by 1 and then we're going to get 5 times negative 1 that's actually a negative 5 then we get t to the negative 2 because again we lower the power by negative 1 trying to resemble the original formula we're going to get 48 t cubed we're going to get minus 3 over the square root of t and then we're going to get minus 5 over t squared which is then the derivative here of p and so I mentioned I mentioned this example that this is some type of linear combination of power functions what do you mean by linear combination a linear combination is when you take you you take coefficients you can add coefficients in front of a function and then you can take addition or subtraction so if you have any functions you can take like 3 times f of x plus 2 times g of x maybe like minus 7 times h of x this is what's often referred to as a linear combination of the function a linear combination why linear here well a linear combination to suggest that when we combine the functions together the only operations we're using is addition subtraction and multiplication by a constant this is exactly how we build linear functions so if you like take the linear equation ax plus by equals c this would be the the equation of a line you just take the linear combination of the variables x and y in three dimensions if you want to make a plane which is like the three-dimensional analog of a line you take ax plus by plus cz oops cz is equal to d so a linear combination of your three variables x y and z forms a plane and so because of that the properties we're exploring in this video are often called the linear properties of the derivative because a derivative you it preserves constant multiples it preserves addition and subtraction therefore when you take the derivative of any linear combination you can always defer it to taking the derivative of the operator of the summands in that linear combination and so these are called the linearity properties of the derivative we're going to find out there are a lot of linear operations in calculus the derivative just being one of them what if we want to take the derivative of f of x equals x cubed plus three times the square root of x over x well if we want to use these linear properties like the the scale of multiples the addition subtraction and the power rule we're not quite there but it turns out that with the appropriate algebraic manipulation we can turn this what looks like a rational expression involving powers and square roots we can actually turn this into a linear combination of power functions and the idea is the following well since you have a monomial in the denominator we can actually break up the fraction into two fractions so we get x cubed over x and then we get three times the square root of x over x but the square root of x we could also write as a power function because when you look at the first expression x cube divided by x I can simplify that just to be x squared I would love to do that with the second one which if I recognize that the square root of x is the one half power then in fact you can do that you take the one half power subtract from that the first power of the denominator will get x squared plus three times x to negative one half we now see like the previous example f of x is in fact a linear combination of these power functions a polynomial is just a linear combination of power functions where the powers are necessarily non-negative integers but taking the derivative of these other linear combinations power functions is just as efficient by the power rule we'll take the derivative of x squared separately which will be 2x then by these rules we'll take the derivative of the next one we'll get three times negative one half times x to the negative three halves power subtract one from negative one half you get negative three halves combining these coefficients together we get 2x we're going to get a minus three halves times x to negative three halves which you could leave that as the derivative that's perfectly fine if you want to throw back in if you want to throw back in you know square roots and fractions you can do that it really doesn't make much of a difference but you can do it it's more just a cosmetic thing 2x minus three over two times x times the square root of x you can do something like that that's appropriate or you could or you could write that last term as the square root of x cubed there's again a couple different ways you could write this all of them cosmetic at this moment what I mostly would care about as an instructor is whether we can do these derivative calculations correct or not can we get from here to here and can we also take the original expression and write it in a form that's more appropriate for the derivative can we prep it for surgery in a manner speaking how about this next one f of x is equal to 4x squared minus 3x quantity squared you're going to see there's a slight issue here that since this is squared we don't have a power function a power function would look like x squared right but we have something more complicated than a power function in fact we could factor this as like u squared composed with 4x squared minus 3x we put a polynomial inside of a power function which doesn't actually make it a power function anymore what we have to do instead to prep it for surgery to prep it for differentiation we need to foil this thing out 4x squared minus 3x times 4x squared minus 3x if we foil this thing out we'll get 4x squared times itself which is 16x to the fourth we'll get 4x squared times a negative 3x which is a negative 12x cubed we'll get that again and for the last term we'll take negative 3x times a negative 3x which will be a positive 9x squared combining like terms we end up with 16x to the fourth minus 24x cubed plus a 9x squared so now we're we're prepped for surgery that is we're prepped for the differentiation taking the derivative here we're going to get 16 times 4 which is 64 rate lower the power of x by 1 so it becomes 64x cubed then for the next one we're going to take negative 24 times 3 which is negative 72 lower the power by 1 to get x squared and then lastly we're going to take 9 times 2 which is 18 lower the power by 1 gives you an x which then gives us the correct derivative of our function here for one last example let's consider finding some tangent lines here of the polynomial function y equals x to the fourth minus 6x squared plus 4 but we want to find the tangent lines for which are horizontal as you can see illustrated here on the graph if your tangent line is horizontal what that means is we're looking for when the derivative is equal to zero so if we calculate the derivative y prime here we're going to get 4x cubed minus 12x and then when you take the derivative of of course the 4 it goes to zero and we want to figure out when this thing equals to zero so we have a polynomial equation we want to solve here that's going to solve it by factoring on the left hand side I notice I can take out the common divisor of 4x that's going to be the greatest common divisor that leaves behind x squared minus 3 in which case then if you take the first one 4x equals 0 that implies x equals 0 if we take x squared minus 3 equals 0 that means x squared equals 3 or x equals plus or minus the square of 3 and so we're going to we see that putting these together there are three places where the function has a horizontal tangent line at 0 at the square of 3 and at the negative square to 3 as we can see here illustrated on the graph