 Hi, I'm Zor. Welcome to Unisor education. We have been introduced to a concept of differential of the function in the previous lecture and right now I would like to talk about the properties of differential. Now this lecture is part of the advanced course of mathematics presented on Unisor.com. The site has not only this particular information about theory of mathematics, but also lots of practical problems with solutions and also has educational functionality like in rolling for instance and taking exam. The site is free, so obviously I encourage you to go to this website and see this video from there. In addition, every lecture has very detailed notes. Okay, so properties of differential. Well, obviously since differential is just a infinitesimal variable which is proportional to differential of the argument with the proportionality coefficient being a derivative. Obviously all the properties of derivatives are applicable very much without change to the properties of differential, right? So, let me just repeat again the definition of differential of the function. It's its derivative at point x times a differential of the argument, which is actually an increment which is converging to zero. Obviously, this is a constant for a constant x, so this one is also converging to zero and this is the coefficient proportion, a proportionality between this and this, which by the way justifies the notation offered by Leibniz, which looks like this for a derivative. All right, so let's just talk about properties of differential. Our first example is if my function is a linear combination of two other functions. That's multiplication. a and b are constant, g and h are two functions. Obviously, they're supposed to be smooth enough. So all the derivatives and differentials exist, etc. So what is differential of f of x? Well, by definition, it's this, right? So the derivative is obviously equals to this, because derivative is additive function and multiplied by dx, right? So that's what derivative of this linear combination is, linear combination of derivatives, which in turn obviously equals to a derivative of g times dx would be dg of x plus b h derivative times dx would be dh of x. So as we see, differential is actually as linear as derivative. That's my first property. Next is product. So f of x equals g of x times h of x. Okay. Differential of f of x is equal to derivative of the product, which is derivative of the first times second plus first derivative of the second, right? That's the derivative of the product times dx, which means it's equal to g derivative and dx is dg of x times h of x plus g of x times h derivative and x is dh of x. So as we see, it looks very much like the derivative of the product. So it's differential of the first times second plus second plus first times differential of the second. Next. Next is reciprocal. Okay. Differential of f is equal to derivative of 1 over g of x. Well, you can actually consider this as a compound function. One function is 1 over x and another function is g of x. And if it's a compound function that you have to really use the chain rule and the chain rule is the derivative of 1 over x is 1 over minus 1 over x square, but instead of x, we go with a g square of x times derivative of the inner function and then times dx, which is equal to this times this is differential of x divided by g square of x. So that's the final answer. Again, obviously you expect the answer like this because you know what would be the derivative. So all these properties of differentials are exact copies of corresponding properties of the derivative. Now what else? Compound function. f of x is equal to g of h of x. Well, we just touched a little bit of compound function when we were talking about reciprocal functions. So let's do it again. d f of x is equal to derivative of this is equal to well, it's g, I will use y here assuming that y is equal to h of x, right? times derivative of the inner function and y is equal to h of x. That's what combination of these two functions actually is compound function. Now I have to multiply it by dx, right? So what is the result? Well the result is g of y times and this is differential of h of x where y is equal to h of x. Here. Well here it probably makes sense to exemplify it somehow. Here is for instance an example which I'm using. For instance g of x is equal to sin of x and h of x is equal to logarithm of x. So we are talking about function f of x equals to sin of logarithm of x. Well let's do first directly without this formula. Now direct calculation of the derivative of the differential in this case is derivative of this function so we know the derivative is equal to derivative of the first one with an argument being the second which is cosine of logarithm of x times the derivative of the second one which is 1 over x and times dx. So that's my direct calculations using definition of the differential. Okay, now how does this formula looks? How does this formula look? Now g is sin so we have a derivative which is from the sin it's cosine of y y is equal to h of x which is logarithm of x times differential of this function logarithm. The differential of logarithm is its derivative which is 1 over x times dx. So we have exactly the same thing. We expect it obviously. Okay. Now based on this we can derive differentials of the functions which are implicitly defined. Let me just make an example, a very simple example. Let's just assume for a second that we don't know what is the derivative of logarithm x. I would like to derive it using implicit methodology. So what can I say about this? That I know the definition of the logarithm is e to the power of logarithm x is equal to x. That's the definition of the logarithm. It's a power which I have to raise the base of this logarithm which is number e to get the argument of the logarithm. Now this I can actually consider as a compound function. So f of x is equal to e to the power of logarithm x. So my g of x is equal to e power x and h of x is equal to logarithm x. So the combination of these two, this one is an inner function and this is an outer function, gives me this. So let's just use the rules for compound functions. So if these are equal to each other then that means their differentials are supposed to be equal to each other. Well on the right I have dx obviously, right? So what's on the left? What is my differential of this function? Well differential of this function is equal to the derivative of the outer function with an argument being the inner function which is logarithm x. Derivative of e to the power of x is e to the power of x, right? Instead of argument x I have to substitute my inner function. And then differential of logarithm x. And this is supposed to be equal to dx, right? Now what is this? e to the power of logarithm x is x, right? That's the definition of the logarithm from which follows that differential of logarithm x is equal to dx divided by x. Which is exactly the same thing as if we will do a derivative of logarithm x which we kind of pretended that we don't know. But actually it is 1 over x times differential of x. So it's exactly the same thing. So that's how we can find the differential implicitly using the compound functions. So this is basically a very simple lecture because all the properties of the differential are mirror limits of properties of derivative because differential is a derivative times dx, basically. That's it. So I do recommend you to read the notes for this lecture on Unisor.com. And other than that, that's it. Thanks very much. Good luck.