CPM Homework Help : APCALC Problem 6-142

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

Hint:

$e^{\ln(x)} = x$ 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.

Example:

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

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

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

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

Step 1:

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

Step 2:

To find $g^\prime(x)$ , take the reciprocal of $f '(x)$ , 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 $f(x) = e^x$ .