# What is the derivative of (ln x)^3?

## What is the derivative of (ln x)^3?

Jan 25, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {\left(\ln \left(x\right)\right)}^{2}}{x}$

#### Explanation:

Use the chain rule for this derivative:

$\frac{d}{\mathrm{dx}} \left({\left(\ln \left(x\right)\right)}^{3}\right) = 3 {\left(\ln \left(x\right)\right)}^{2} \cdot \frac{d}{\mathrm{dx}} \left(\ln \left(x\right)\right) = \frac{3 {\left(\ln \left(x\right)\right)}^{2}}{x}$.

Alternatively $y = {u}^{3}$ and $u = \ln \left(x\right)$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {u}^{2} \cdot \frac{1}{x}$ and substituting $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {\left(\ln \left(x\right)\right)}^{2}}{x}$