# Derivative of x^x: Formula, Proof by First Principle

The derivative of xx (x to the power x) is equal to xx(1+logex). In this post, we will learn the formula for the derivative of xx 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.

## Derivative of xx Formula

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

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

where ln denotes the natural logarithm (log with base e), that is, lnx=logex.

## Derivative of xx 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=xx

Taking logarithms with base e both sides, we obtain that

loge y = loge xx

⇒ loge y = xloge x by the logarithm rule logabk = k logab.

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

d/dx(loge y) = d/dx(xloge x)

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

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

⇒ $\dfrac{dy}{dx}$ = y(1 + logex)

⇒ $\dfrac{dy}{dx}$ = xx(1 + logex) as y=xx.

So the derivative of xx (x to the x) is equal to xx(1 + logex) and this is obtained by the logarithmic differentiation.

## Derivative of xx 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)=xx in the above formula (I). Thus we have:

Thus, the derivative of xx is xx(1+logex) and this is obtained by the first principle of derivatives, that is, by the limit definition of derivatives.

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Discontinuity of a Function

Derivative: Definition, Formulas, Examples

Integration: Definition, Formulas, Examples

## FAQs on Derivative of xx

Q1: What is the derivative of xx?

Answer: The derivative of xx (x to the x) is xx(1+logex).

Q2: What is the derivative of 2x?

Answer: The derivative of 2x is 2xloge2.

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