Tricky question regarding Inverse functions

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1. Mar 24, 2016

Geologist180

1. The problem statement, all variables and given/known data
1. Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ?
If it helps the answer is (-5/32)

2. Relevant equations

f-1'(b)=1/(f')(a)

3. The attempt at a solution

Im not really sure how to start this problem. I am familiar with how to use the equation above to plug solve for "c" and then plug c into the derivative, so I assume it has something to do with that but I cant find any helpful resources on the web showing me how to do this specific type of problem.

2. Mar 24, 2016

LCKurtz

I suppose you mean G'(2)

Start by showing us the calculation for G' using the chain rule.

3. Mar 24, 2016

Geologist180

I believe the derivative of G would be -1/(f-1) 2
Is this correct?

4. Mar 24, 2016

LCKurtz

No. You haven't used the chain rule on the definition of G.

5. Mar 24, 2016

Geologist180

Im confused how I am supposed to use the chain rule in the context of this question.
When I think about the chain rule, I think about more complicated functions and taking the derivative of the functions with respect to the outside of the brackets multiplied by the derivative of the inside of the brackets. Im having trouble seeing how to apply that here.

6. Mar 24, 2016

LCKurtz

Isn't that exactly what you have:
$$G(\cdot)=(f^{-1}(\cdot))^{-1}$$where you can replace the dot with whatever variable you want to call it, maybe $y$ if you call the original equation $y=f(x)$. What is the derivative of the "inside"?

I know it is a bit confusing with one of the $-1$'s being a negative exponent and the other an inverse function.

7. Mar 24, 2016

Geologist180

I believe it would be: (substituted a star for the dot)

( - (f-1(*))-2) (d/dx (f-1(*)))

Am I following you correctly in this?
Thank you for your hep by the way

8. Mar 24, 2016

LCKurtz

With the * that would be written ( - (f-1(*))-2) (d/d* (f-1(*))). Let's use $y$ and write it$$G'(y)=-(f^{-1}(y))^{-2}\frac d {dy} f^{-1}(y)$$Now, remember your question is asking for $G'(2)$. If you put that in you have$$G'(2) =-(f^{-1}(2))^{-2}\frac d {dy} f^{-1}(2)$$So now you have to ask yourself: do I know $f^{-1}(2)$? Do I know the derivative of the inverse function at $2$? What do I know about derivatives of inverses versus derivatives of the original function?