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

## What is the Derivative of x^{lnx}?

**Answer:**

**Explanation:**

To find the derivative of x^{lnx}, we will use the logarithmic differentiation. Let us put

y = x^{lnx}.

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

ln y = ln x^{lnx}

⇒ 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}

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

So the derivative of x^{lnx} is equal to 2x^{lnx -1} lnx.

**More Derivatives:** Derivative of 1/lnx

## FAQs

### Q1: What is the differentiation of x^{lnx}?

**Answer:** The differentiation of x^{lnx} is equal to 2x^{lnx -1} lnx.