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DerivativesCalculusCollegeabout 0:49

Differentiate x^x: animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (x)^(x)·(ln(x) + 1).

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Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[{x}^{x}\right]$
2. Step by step
1. $\frac{d}{dx}\left[{x}^{x}\right] = {x}^{x} \left(\ln\left( x\right)+1\right)$
  Logarithmic differentiation: d/dx[uᵛ] = uᵛ(v′ ln u + v u′/u)
2. $f'(x) = {x}^{x} \left(\ln\left( x\right)+1\right)$
  Combine and simplify
3. The answer
$f'(x) = {x}^{x} \left(\ln\left( x\right)+1\right)$
f'(x) = (x)^(x)·(ln(x) + 1)
4. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: differentiate x to the power x. Find the derivative of x to the power x with respect to x.

Step by step. Logarithmic differentiation: d by d x[uᵛ] equals uᵛ(v prime ln u plus v u prime /u). Combine and simplify.

The answer. So the answer is f prime of x equals (x) to the power x times (ln(x) plus 1). The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

Check. Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.

About derivatives

Differentiate with the power, product, quotient and chain rules, including higher derivatives.

Differentiate x^x: Derivatives Step-by-Step Lesson