### Home > APCALC > Chapter 6 > Lesson 6.4.2 > Problem6-142

6-142.

Recall that $e^{\ln(x)} = x$. Use the formula for the derivative of an inverse to compute $\frac { d } { d x }\ln(x)$. Homework Help ✎

$e^{\ln(x)} = x$ is another way of saying that $y = e^x$ and $y = \ln(x)$ are inverse functions.
Inverse functions have reciprocal derivatives at their corresponding $(x, y) → (y, x)$-values.

If $f(x)$has the coordinate point $(a, b)$,

$f'(a)=\frac{2}{5},$

and $g(x)$ is the inverse of $f(x)$, then

$g'(b)=\frac{5}{2}.$

Let $f(x) = e^x$ and $g(x) = \ln(x)$.
First find $f ^\prime(x) =$ _____.

To find $g^\prime(x)$, take the reciprocal of$f '(x)$, but NOT at the
x-value.

$g'(x)=\frac{1}{f'(y)}=\frac{1}{f'(g(x))}$

Now evaluate $g^\prime(x)$ when $f(x) = e^x$.