Calculus Chain Rule of composite functions

Question

f and g are functions satisfying f'(0)=g(1)=g'(1)=1. Let h(x)=f(g(x^2)+x) find h'(1) ?

Answer 1

Hi Dav!

The chain rule has that the derivative of a composite function is a product (or "chain") of the derivative of the outer function and the derivative of all functions inside. Think of these like layers of an onion, and peel them away.

Let's solve this using x's, and only put in x=1 at the end. We begin:

h'(x) = f'(g(x^2)+x)*[g'(x^2)*2x+1]

Notice that I took the derivative of the f, and then multiplied this by the derivative of its argument. I did this again in the case of g(x^2).

Now substitute x=1. It doesn't look to me like you have all the pieces of the puzzle; maybe you misremembered the problem? (If your g(x^2)+x is actually g(x^2)-x, that would mean you have all you need.) If you do have the function and derivative values you need, substitute these, and you're done! And if not, you can use whatever you have to provide the best answer you can for h'(1).

Cheers,

Joe