Chain Rule f(x)=(x+1)3 Hey everyone. We’re going to do a couple of chain rule problems. And chain rule is a really basic concept that can apply to almost any kind of math problem. It’s an operation like multiplication or division or what a square root means. So, it’s just kind of something fundamental that you’ll need to know so that you can apply it whenever it’s relevant. We’re just going to learn about it with some really basic problems here by taking some derivatives. So, the first problem we’re going to do, f(x)=(x+1)3, and we’re going to go ahead and take the derivative of this problem to illustrate the cons of the chain rule. So, I could put this in the derivative section but we’re going to kind of ignore that part just of the chain rule. So the derivative, of course f1(x), and then chain rule basically what it means is it allows you to deal with something that’s a little more complex. We have two terms here inside of this parenthesis and then an exponent that’s applying to the term as a whole. You can use chain rule to take the derivative of this function without having to multiply everything out. So, the way that we’re going to do that is we know before, for example, if we were going to take the derivative of x3, the derivative of x3 we know to be 3x2, we bring the exponent in front here to the three, and then we subtract one from the exponent and we get two. So this is the derivative. Similarly, with chain rule, we can bring the exponent in front here to be the coefficient. So we’re going to say, 3x+1 and then we subtract one from the exponent and we have two there. So, we can do this except that x+1, there are two terms in here instead of just an x like we had up here. So we can’t just bring the three out in front, make this a two and be done. We have to deal with the inside and make sure that we’ve cover all of our basis. So that’s where chain rule comes in. Chain rule says do a part on the outside, bring the three over and make this a two but then deal with the inside. So all we do, when we take the derivative of anything and we’re using chain rule, we go ahead and perform this operation. We’ll bring the three over and make this a two. But then, we multiply what we have by the derivative of the inside. So, we take the derivative of the outside, and then multiply by the derivative of the inside. So, the derivative of x+1, the derivative of x is simply 1, and then the derivative of 1 is zero, of course it’s a constant that goes away. So, the derivative here is just one. So in this case, we would multiply this function by one, the derivative of the inside here and that would be your answer. So, it actually ends up being the same thing, 3*x+12. And the chain rule doesn’t have any effect on the final answer because it’s just one, the derivative of the inside is just one and it gets absorbed. But we’ll see in the next example. Now that we’ve illustrated the basic concept, what it looks like when the derivative of the inside actually affects the optimum of the problem. Okay? Thanks.