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

##### 1 Answer
Dec 20, 2015

${\left(x + 1\right)}^{x} \left(\ln \left(x + 1\right) + \frac{x}{x + 1}\right)$

#### Explanation:

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

Take the natural log of both sides.

$\ln y = \ln {\left(x + 1\right)}^{x}$

Simplify using logarithm rules.

$\ln y = x \ln \left(x + 1\right)$

Take the derivative of both sides.

$\frac{y '}{y} = \ln \left(x + 1\right) + \frac{x}{x + 1}$

Multiply both sides by $y$, which equals ${\left(x + 1\right)}^{x}$.

$y ' = {\left(x + 1\right)}^{x} \left(\ln \left(x + 1\right) + \frac{x}{x + 1}\right)$