Question

Suppose f and g are functions differentiable on (0, +infinity). If \[f(2\sqrt{x})=x^2g(\sqrt{x}+2)\] and it is known that f'(4)=7 and g'(4)=-3, determine g(4).

Answer 1

Hey there :)

Differentiate each side with respect to x.
You'll need to do some product rule business on the right side,
and then a bunch of chain rule after that, ya?

Since you're applying product rule, you should end up with both g and g',
so you'll be able to solve for g with some algebra.

Answer 2

Hi! And thank you! :)

Answer 3

\[\large\rm f(2\sqrt{x})=x^2g(\sqrt{x}+2)\]Differentiating,\[\large\rm f'(2\sqrt{x})\cdot\color{royalblue}{(2\sqrt x)'}=\color{royalblue}{(x^2)'}g(\sqrt{x}+2)+x^2\color{royalblue}{(g(\sqrt{x}+2))'}\]So what I've done here is,
Applied chain rule to the left side,
and only "set up" the product rule on the right so far.

Answer 4

Confusing? :o
What do you think?

Answer 5

It's not confusing so far. I think I can figure out the rest.

Answer 6

cool :)