Question

The functions f and g are differentiable and f(g(x))=x^2 for all x. If f(4)=8, g(4)=8, f'(8)=-2, what is the value of g'(4)?

Answer 1

f(g(x))=x^2


f'(g(x))*g'(x)=2x


f'(g(4))*g'(4)=2(4)


Do you see what to do from here?

Answer 2

What should I do next?

Answer 3

g(4) = ??

Answer 4

I don't get it.

Answer 5

what is g(4)

Answer 6

8.

Answer 7

so

f'(g(4))*g'(4)=2(4)

becomes

f'(8)*g'(4)=2(4)

Answer 8

what is f'(8)

Answer 9

-2.

Answer 10

now replace f'(8) with -2 and solve for g'(4)

Answer 11

So the answer is -2?

Answer 12

yes

Answer 13

Thanks for the help. Sorry I was clueless.