# How do you find the derivative of ln(1+e^x)?

Apr 24, 2016

$f ' \left(x\right) = {e}^{x} / \left(1 + {e}^{x}\right)$

#### Explanation:

The chain rule says that if

$f \left(x\right) = g \left(h \left(x\right)\right)$

then

$f ' \left(x\right) = h ' \left(x\right) g ' \left(h\right)$

Substituting in the functions from the question,

$h \left(x\right) = 1 + {e}^{x}$

$g \left(h\right) = \ln \left(h\right)$

Then the derivatives are

$h ' \left(x\right) = {e}^{x}$

$g ' \left(h\right) = \frac{1}{h}$

According to the chain rule,

$f ' \left(x\right) = \left({e}^{x}\right) \cdot \left(\frac{1}{h \left(x\right)}\right)$

Expanding out and substituting in that $h \left(x\right) = 1 + {e}^{x}$,

$f ' \left(x\right) = {e}^{x} / \left(1 + {e}^{x}\right)$