# What is the derivative of ln(1/x)?

Sep 1, 2016

The derivative is $- \frac{1}{x}$

#### Explanation:

Here are two ways to find the derivative:

Method 1 relies on knowing about exponents and logarithms.

We'll use $\frac{1}{x} = {x}^{-} 1$ and $\ln \left({a}^{b}\right) = b \ln a$ and $\frac{d}{\mathrm{dx}} \left(c f \left(x\right)\right) = c \frac{d}{\mathrm{dx}} \left(f \left(x\right)\right)$

We get:

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

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

$= - \frac{1}{x}$.

Method 2 uses the chain rule.

$\frac{d}{\mathrm{dx}} \left(\ln u\right) = \frac{1}{u} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$ and

$\frac{d}{\mathrm{dx}} \left(\frac{1}{x}\right) = \frac{d}{\mathrm{dx}} \left({x}^{-} 1\right) = - {x}^{-} 2 = - \frac{1}{x} ^ 2$.

We get

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

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

$= - \frac{1}{x}$