# What is the second derivative of f(x)= ln (x^3+e^x)?

Feb 4, 2016

$f ' ' \left(x\right) = \frac{\left({x}^{3} + {e}^{x}\right) \left(6 x + {e}^{x}\right) - {\left(3 {x}^{2} + {e}^{x}\right)}^{2}}{{x}^{3} + {e}^{x}} ^ 2$

#### Explanation:

We first use the log rule to find

$f ' \left(x\right) = \frac{3 {x}^{2} + {e}^{x}}{{x}^{3} + {e}^{x}}$

We now use the quotient rule to find

$f ' ' \left(x\right) = \frac{\left({x}^{3} + {e}^{x}\right) \left(6 x + {e}^{x}\right) - \left(3 {x}^{2} + {e}^{x}\right) \left(3 {x}^{2} + {e}^{x}\right)}{{x}^{3} + {e}^{x}} ^ 2$