# What is the second derivative of f(x)= ln (x+e^x)?

Jan 14, 2016

$f ' ' \left(x\right) = \frac{\left(x - 2\right) {e}^{x} - 1}{x + {e}^{x}} ^ 2$

#### Explanation:

From the standard rules of differentiating log functions we get

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

From the quotient rule we now get

$f ' ' \left(x\right) = \frac{\left(x + {e}^{x}\right) \cdot {e}^{x} - \left(1 + {e}^{x}\right) \cdot \left(1 + {e}^{x}\right)}{x + {e}^{x}} ^ 2$

$= \frac{\left(x - 2\right) {e}^{x} - 1}{x + {e}^{x}} ^ 2$.