The derivative of x^{x} (x to the power x) is equal to x^{x}(1+log_{e}x). In this post, we will learn the formula for the derivative of x^{x} and how to find it. To calculate the derivative of x to the x, we will use the following methods:

- Logarithmic differentiation
- First principle of derivatives.

Table of Contents

## Derivative of x^{x} Formula

The derivative of x^{x} is denoted by $\dfrac{d}{dx}$(x^{x}) or (x^{x})$’$. The formula of the derivative of x^{x} is given as follows.

$\dfrac{d}{dx}$(x^{x}) = x^{x}(1+lnx),

where ln denotes the natural logarithm (log with base e), that is, lnx=log_{e}x.

## Derivative of x^{x} by Logarithmic Differentiation

Note that we use the logarithmic differentiation method to find the derivative of a function having another function as an exponent. Thus, we can find the derivative of x to the power x using this method. Let us put

y=x^{x}

Taking logarithms with base e both sides, we obtain that

log_{e} y = log_{e} x^{x}

⇒ log_{e} y = xlog_{e} x by the logarithm rule log_{a}b^{k} = k log_{a}b.

Differentiating both sides w.r.t. x, we get that

d/dx(log_{e} y) = d/dx(xlog_{e} x)

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}$ = x d/dx(log_{e} x) + log_{e}x d/dx(x) by the product rule of derivatives.

⇒ $\dfrac{1}{y} \dfrac{dy}{dx}$ = x ⋅ 1/x + log_{e}x ⋅ 1

⇒ $\dfrac{dy}{dx}$ = y(1 + log_{e}x)

⇒ $\dfrac{dy}{dx}$ = x^{x}(1 + log_{e}x) as y=x^{x}.

So the derivative of x^{x} (x to the x) is equal to x^{x}(1 + log_{e}x) and this is obtained by the logarithmic differentiation.

## Derivative of x^{x} by First Principle

The derivative of f(x) by the first principle, that is, by the limit definition is given by

$\dfrac{d}{dx}(f(x))$$=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$ $\cdots$ **(I)**

We will use the following fact:

$\lim\limits_{h\to 0}\dfrac{x^h-1}{h}=y$ if and only if $x=\lim\limits_{n\to\infty}\left(1+\dfrac yn\right)^n$ if and only if $x=e^y \iff y=\log(x)$

Put f(x)=x^{x} in the above formula **(I)**. Thus we have:

Thus, the derivative of x^{x} is x^{x}(1+log_{e}x) and this is obtained by the first principle of derivatives, that is, by the limit definition of derivatives.

**Must Read:**

**Limit: Definition, Formulas, Examples**

**Derivative: Definition, Formulas, Examples**

**Integration: Definition, Formulas, Examples**

## FAQs on Derivative of x^{x}

**Q1: What is the derivative of x**

^{x}?Answer: The derivative of x^{x} (x to the x) is x^{x}(1+log_{e}x).

**Q2: What is the derivative of 2**

^{x}?Answer: The derivative of 2^{x} is 2^{x}log_{e}2.