Chain Rule for Derivatives

Chain Rule for Derivatives Let's discuss the chain rule, which is really nothing more thanthe derivative of a composite function.What is the derivative of the sine of 3x?It is common to assume that the derivative of sine x is cosine x, because the inside part of the function does not change. However, we will find that this is incorrect. In the second example,it is setted up as the derivative of sin x with respect to x and its amplitude. This will verify thehypothesis that cos x = 3 sin x. The amplitude is different by afactor of 3. This tells us that the derivative is not cos x.Rather, it is 3 cos x with a coefficient of 3 in front. This comes from the derivative of 3x, a quantity which is now part of theanswer itself.

So if you take the derivative of a composite function, we'll callit f of g of x, what you'll do is you'll take the derivative. Wetook the derivative of sine, we said it was cosine. So in this case, since f is already a function, you are going to use thechain rule like this. And then this 3x remained exactly the same, so I will still have gof x. But then what you do is you look at this inner component right here, and you will take the derivative of that as well. Thederivative of g of x is g prime of x. So this is one notation forthe chain rule. You take the derivative of some function of u,which would be equal to f prime of u times you might either see it as u prime or du, asin derivative of u. The derivative of f (u) = (du/dx)(du/dx) canbe written as the derivative of u (x) with respect to x plus thederivative of u with respect to x times the derivative of f (u) with respect to u.