Let f be a differentiable function with f(3) = 5 and f'(3) = 2. If g is the inverse function of f, what is the value of g'(5)?

Step-by-step solution

1. 1

   Start from the defining property of an inverse, not from a formula you half-remember. By definition $g$ undoes $f$, so for every $x$ in the domain

   $f\bigl(g(x)\bigr)=x.$

   Everything else follows from differentiating this identity.

2. 2

   Differentiate both sides with respect to x using the chain rule. The left side is a composition, the right side has derivative $1$:

   $f'\bigl(g(x)\bigr)\cdot g'(x)=1\quad\Longrightarrow\quad g'(x)=\frac{1}{f'\bigl(g(x)\bigr)}.$

3. 3

   Find the inner value g(5) before substituting. This is the step students skip. Since $f(3)=5$, applying $g$ to both sides gives

   $g(5)=3.$

   So the derivative of $f$ has to be read at $3$, not at $5$.

4. 4

   Substitute the two known numbers.

   $g'(5)=\frac{1}{f'(g(5))}=\frac{1}{f'(3)}=\frac{1}{2}.$

5. 5

   Interpret the result. The graphs of $f$ and $g$ are reflections in the line $y=x$, and reflecting swaps rise with run — so slopes become reciprocals. A function climbing with slope $2$ at $(3,5)$ has an inverse climbing with slope $\tfrac12$ at $(5,3)$.