 One of the most powerful extensions of the derivative rules is known as the chain rule. This is a true power tool of mathematics, and like all power tools, unless you use it carefully, you can make a real mess of things really quickly. So the chain rule appears in connection with composite functions. Suppose f of x is p of q of x. What is f prime of x? And the answer to this question is given by the chain rule. So if you want to find the derivative of a composite function, it's going to be the derivative of the outside function, p prime, evaluated at the inside function, q of x, times the derivative of the inside function, q prime of x. Now, this formula is wonderfully compact and terribly misleading, so let's do a few examples. So let's find the derivative of f of x equals 2x plus 7 quantity squared, and since the chain rule is a power tool of mathematics, let's make sure we're using it correctly. So the chain rule tells us that the derivative of a composite function is the derivative of the outside function, evaluated at the inside function, times the derivative of the inside function. Or a better way of looking at this is this is the derivative of the last thing that you do, evaluated at the first thing you did, times the derivative of that first thing. And so this leads to the following approach to evaluating the derivative of a composite function. First, remember that the type of function is always determined by the last operation performed. And so for the first part of the chain rule, we can ignore or drop out that inside function completely, then find the derivative of what we see, and then apply the kindergarten rule, which is put things back where you found them. We took the Q of x out, we should put it back in, and also multiply by the derivative of the thing that we just wrote. Okay, so let's apply that. Our function is a squaring function. So what we'll do is we'll ignore everything except for the squaring portion, and so our function is now squared. And now we want to find the derivative of squared. And the important thing is that the chain rule says just differentiate that normally, but then multiply by the derivative of, which will be 2 to the first, times the derivative of. And now we invoke our kindergarten rule, put things back where you found them. Originally in this, we had 2x plus 7, so every place we see will write 2x plus 7. Now this does leave us with some unfinished business. We have the derivative of 2x plus 7, so we find that derivative and simplify the algebra to get our derivative according to the chain rule. Now it'd be great if this were the actual derivative, but let's check by finding the derivative using another method. And here we might use some algebra and note that 2x plus 7 squared is the same as 2x plus 7 times 2x plus 7. And this is a product, so we'll apply the product rule and find the derivative. But this is different from the form that we got using the chain rule, so we'll have to do some algebra to see if they are really the same thing. And we see that after all the dust settles, the derivative we get from using the chain rule and the derivative we get that we know is correct using the product rule are the same. And so we have some confidence that our use of the chain rule is correct. How about the derivative of something else? So first of all, it'll be convenient to write this in exponential notation. And since the last operation performed is a cube root, this is a cube root function. And again, for purposes of finding the derivative using the chain rule, it's convenient to ignore all but the last thing that we do. So instead of thinking about this as x squared plus 4x plus 7 raised to power 1 third, what we can do is we can drop out everything except for the last part, raised to power 1 third, and think of our function as tz to the 1 third power. So what's the derivative of tz to the 1 third power? That'll be 1 third tz to the 2 thirds power times the derivative of tz. And we invoke the kindergarten rule, put things back where you found them. And we still have a derivative we need to do, so we'll find that value of that derivative. And everything we do after this point is in the manner of algebraic cleanup.