Chain Rule for Derivatives
Let’s discuss the chain rule, which is really nothing more than the 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. [Graph] In the second example,it is setted up as the derivative of sin x with respect to x and its amplitude. This will verify the x. hypothesis that cos x – 3 sin x. The amplitude is different by a factor 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 the answer itself. [Graph] So if you take the derivative of a composite function, we’ll call it fof g of x, what you’ll do is you’ll take the derivative. We took the derivative of sine, we said it was cosine. So in this case, since f is already a function, you are going to use the chain rule like this. And then this 3x remained exactly the same, so I will still have g of 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. The derivative of g of x is g prime of x. So this is one notation for the 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, as in derivative of u. The derivative of f (u) – (du/dx)(du/dx) can be written as the derivative of u (x) with respect to x plus the derivative of u with respect to x times the derivative off (u) with respect to u. Chain Rule d/dx sin(3x)=cos(3x) d/dx 3x=3=3cos(3x) d/dx f(g(x))=f'(g(x))*g'(x) d/dx f(u)=f'(u)*du=f'(u)*u’ d/dx f(u(x))=(df/du)*(du/dx)
