F(2)=9, f’(2)=4, f(5)=4, f’(5)=9, g(2)=1, g’(2)=-3 and H(x)=f(x^2+g(x)) Find dH/dx|x=2

Answer 1

Hi there Joanne!

This is all just a problem of "plug and chug." The goal is to find dH/dx | x=2 (the derivative of H with respect to x at x = 2). The confusing part may be that we don't have any functions to work with, but that's okay. The things they give you are even clues in and of themselves - look in those given values on hard questions like this for clues on where you might try to start.

The way to start this problem is by plugging making x=2 and solve for H(2). Let's not take the derivative of H just yet.

H(x) = f(x^2 + g(x)) ; f is outside the parantheses, so we don't know what to take f of just yet.

H(2) = f(2^2 + g(2)) ; we are given that g(2) = 1

= f(4 + 1)

H(2) = f(5)

We've now simplified it down. You might have noticed that we're given f'(5) = 9, and can probably tell where this is going. This is where we take the derivative of both sides, thus of H and f.

H'(2) = f'(5)

H'(2) = 9

You do want to watch out for superfluous information. Here, F(2), f'(2), and g'(2) ended up being irrelevant. So, you don't want to lead yourself astray either with false clues, looking at all the givens.

Hope this helps and reaches you in time! Please reach out if you have any further questions.