# What is the derivative of x^(10x)?

## What is the derivative of x^(10x)?

Jan 25, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(10 \ln \left(x\right) + 10\right) {x}^{10 x}$

#### Explanation:

$y = {x}^{10 x}$

Take the logarithm of both sides then use implicit differentiation:

$\ln y = \ln \left({x}^{10 x}\right)$

$\to \ln y = 10 x \ln \left(x\right)$ (Using the laws of logs)

Now implicitly differentiate via the product rule:

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 10 \ln \left(x\right) + 10 x \cdot \frac{1}{x} = 10 \ln \left(x\right) + 10$

Multiply by $y$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(10 \ln \left(x\right) + 10\right) y$

Substitute $y$ back in:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(10 \ln \left(x\right) + 10\right) {x}^{10 x}$