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Find the derivative of x^(1/3): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = x^(-(2/3)) / 3.

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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
$\frac{d}{dx}\left[{x}^{\frac{1}{3}}\right]$
2. Step by step
1. $\frac{d}{dx}\left[{x}^{\frac{1}{3}}\right] = \frac{{x}^{-\frac{2}{3}}}{3}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. $f'(x) = \frac{{x}^{-\frac{2}{3}}}{3}$
  Combine and simplify
3. The answer
$f'(x) = \frac{{x}^{-\frac{2}{3}}}{3}$
f'(x) = x^(-(2/3)) / 3
4. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of x to the power 1 over 3). Find the derivative of x to the power 1 over 3) with respect to x.

Step by step. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Combine and simplify.

The answer. So the answer is f prime of x equals x to the power -(2 over 3)) / 3. 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.

Find the derivative of x^(1/3): Animated Step-by-Step Lesson