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DerivativesCalculusCollegeabout 1:13

Third derivative of x^5: animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'''(x) = 60x².

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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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d^{3}}{dx^{3}}\left[{x}^{5}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[{x}^{5}\right]$
  Derivative number 1
2. $\frac{d}{dx}\left[{x}^{5}\right] = 5{x}^{4}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. $f'(x) = 5{x}^{4}$
  Simplify the first derivative
4. $\frac{d}{dx}\left[5{x}^{4}\right]$
  Derivative number 2
3. Steps, part 2
1. $\frac{d}{dx}\left[5{x}^{4}\right] = 20{x}^{3}$
  Constant multiple rule
2. $\frac{d}{dx}\left[{x}^{4}\right] = 4{x}^{3}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. $f''(x) = 20{x}^{3}$
  Simplify the second derivative
4. $\frac{d}{dx}\left[20{x}^{3}\right]$
  Derivative number 3
4. Steps, part 3
1. $\frac{d}{dx}\left[20{x}^{3}\right] = 60{x}^{2}$
  Constant multiple rule
2. $\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. $f'''(x) = 60{x}^{2}$
  Simplify the third derivative
5. The answer
$f'''(x) = 60{x}^{2}$
f'''(x) = 60x²
6. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: x to the power 5. Find the third derivative of x to the power 5 with respect to x.

Steps, part 1. Derivative number 1. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Simplify the first derivative. Derivative number 2.

Steps, part 2. Constant multiple rule. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Simplify the second derivative. Derivative number 3.

Steps, part 3. Constant multiple rule. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Simplify the third derivative.

The answer. So the answer is f prime prime prime (x) equals 60x squared. 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.

Third derivative of x^5: Derivatives Step-by-Step Lesson