Question

"Easy" Calculus!
Two similar questions:

1) If f(4) = 5 and f'(4) = 2/3, then calculate the derivative of f inverse of 5.

2) If g(7) = 3 and g'(3) = 5/6 and g'(7) = 3/4. then calculate the derivative of f inverse of 3

The answer to #1 is 3/2 and #2 is 4/3, but could someone fill me in on the intermediate steps/general method for solving?

Answer 1

sure

Answer 2

\[\left(f^{-1}(x)\right)'=\frac{1}{f'(f^{-1}(x)}\]

Answer 3

and so
\[\left(f^{-1}(5)\right)'=\frac{1}{f'(f^{-1}(5)}\]

Answer 4

now for your problem
\[f(4)=5\implies f^{-1}(5)=4\]

Answer 5

so
\[\left(f^{-1}(5)\right)'=\frac{1}{f'(f^{-1}(5)}\]\[=\frac{1}{f'(4)}=\frac{1}{\frac{2}{3}}=\frac{3}{2}\]

Answer 6

second one is similar

Answer 7

both these problems were to get you to practice
\[\left(f^{-1}(x)\right)'=\frac{1}{f'(f^{-1}(x)}\]

Answer 8

Okay, so I've see the \[(f '^ {-1}(x))\] equals 1/f' whatever..., but was wondering, when I'm solving for the derivative of f inverse of 5, its ok just to use 1 over f'(4)?

Answer 9

Oh wait. I see why.