# Derivative of x^lnx [x to the power lnx]

The derivative of x^lnx (x to the power lnx) is equal to 2xlnx -1 lnx. Here, ln denotes the natural logarithm, that is, lnx =loge x. In this post, we will learn how to differentiate xlnx.

## What is the Derivative of xlnx?

Explanation:

To find the derivative of xlnx, we will use the logarithmic differentiation. Let us put

y = xlnx.

We need to find dy/dx. Taking logarithms both sides, we get that

ln y = ln xlnx

⇒ ln y = lnx lnx

⇒ ln y = (lnx)2

Differentiating with respect to x, we obtain that

$\dfrac{1}{y} \dfrac{dy}{dx}= 2 \ln x \dfrac{d}{dx}(\ln x)$, by the chain rule of derivatives.

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}= 2 (\ln x) \dfrac{1}{x}$

⇒ $\dfrac{dy}{dx}= 2 y (\ln x)$ x-1

⇒ $\dfrac{dy}{dx}= 2\ln x$xlnx-1 as y=xlnx.

So the derivative of xlnx is equal to 2xlnx -1 lnx.

More Derivatives: Derivative of 1/lnx

Derivative of ln(lnx)

Derivative of sinx/x

Derivative of 1/sin x

## FAQs

### Q1: What is the differentiation of xlnx?

Answer: The differentiation of xlnx is equal to 2xlnx -1 lnx.