CPM Homework Help : CALC Problem 6-149

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

Hint:

$e^{\operatorname{ln}x} = x$ 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.

Example:

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

Step 1:

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

Step 2:

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

Step 3:

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