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

Jun 26, 2017

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

#### Explanation:

$\text{differentiate using the "color(blue)"chain rule}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

$\frac{d}{\mathrm{dx}} \left(\ln \left(f \left(x\right)\right)\right) = \frac{1}{f \left(x\right)} \times f ' \left(x\right)$

$\Rightarrow \frac{d}{\mathrm{dx}} \left(\ln \left(1 + \frac{1}{x}\right)\right) = \frac{1}{1 + \frac{1}{x}} \times \frac{d}{\mathrm{dx}} \left(1 + {x}^{-} 1\right)$

$= \frac{1}{1 + \frac{1}{x}} \times - \frac{1}{x} ^ 2$

$= - \frac{1}{x \left(x + 1\right)}$