### Home > CALC > Chapter 6 > Lesson 6.4.2 > Problem6-149

6-149.

Recall that $e^{\operatorname{ln}x} = x$, use the derivative of the inverse to find $\frac { d } { d x } ( \operatorname { ln } x )$.

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

If $f(x)$ has coordinate point $(a, b)$.
And $f'(a)=\frac{2}{5}$
And if $g(x)$ is the inverse of $f(x)$, then $g'(b)=\frac{5}{2}$.

Let $f(x) = e^x$ and $g(x) =\operatorname{ln}x$.
First find $f^\prime(x) =$ _____. (This should be easy!)

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

$g'(x)=\frac{1}{f'(y)}=\frac{1}{f'(g(x))}$
Now evaluate $g^\prime(x)$ when $f(x) = e^x$.