Derivatives of Inverse Functions

Question

"Suppose g is the inverse function of a differentiable function f, and G(x)=1/g(x). If f(3)=7 and f'(3)=1/9, find G'(9). FInd G'(7)."

 

Please show all steps- thanks so much!!

Answer 1

 

G'(x) = d/dx 1/g(x) = (-g'(x))/(g(x)²)

g'(x) = 1/(f'(g(x)), so

G'(x) = -1/(g(x)²*f'(g(x)))

 

G'(7) = -1/(3²*1/9) = -1

 

You don't have enough information to find G'(9). The inverse is the mirror image across the line x=y, and we don't know at all what the solutions of f(x)=9 are. We need that info to plug into the denominator of G'(x)