# What is the derivative of log_e(x)?

Apr 4, 2018

$\frac{1}{x}$

#### Explanation:

${\log}_{e} \left(x\right)$ is commonly denoted as $\ln \left(x\right)$, the natural log.

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

If you would like a proof, we can derive it from the limit definition:

${\lim}_{\delta x \to 0} \frac{f \left(x + \delta x\right) - f \left(x\right)}{\delta x}$

$= {\lim}_{\delta x \to 0} \frac{\ln \left(x + \delta x\right) - \ln \left(x\right)}{\delta x}$

$= {\lim}_{\delta x \to 0} \frac{\ln \left(\frac{x + \delta x}{x}\right)}{\delta x}$

$= {\lim}_{\delta x \to 0} \frac{1}{\delta x} \ln \left(1 + \frac{\delta x}{x}\right)$

$= {\lim}_{\delta x \to 0} \ln \left({\left(1 + \frac{\delta x}{x}\right)}^{\frac{1}{\delta x}}\right)$

$= {\lim}_{\delta x \to 0} \ln \left({\left(1 + \frac{\delta x}{x}\right)}^{\frac{1}{\delta x}}\right)$

$\text{Let } \tau \equiv \frac{\delta x}{x}$:

$= {\lim}_{\delta \tau \to 0} \ln \left({\left(1 + \tau\right)}^{\frac{1}{x \tau}}\right)$

$= {\lim}_{\delta \tau \to 0} \ln \left[{\left({\left(1 + \tau\right)}^{\frac{1}{\tau}}\right)}^{\frac{1}{x}}\right]$

$= \ln \left[{\left(e\right)}^{\frac{1}{x}}\right]$

$= \frac{1}{x} \ln \left(e\right)$

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

$= \frac{1}{x}$