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DerivativesCalculusCollegeabout 1:14

Find d/dx((x^2 - 1)^3 (x + 2)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = 6(x² - 1)²x(x + 2) + (x² - 1)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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[{\left({x}^{2}-1\right)}^{3} \left( x+2\right)\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[{\left({x}^{2}-1\right)}^{3} \left( x+2\right)\right] = 6 {\left({x}^{2}-1\right)}^{2} x \left( x+2\right)+{\left({x}^{2}-1\right)}^{3}$
  Product rule: (uv)′ = u′v + uv′
2. $\frac{d}{dx}\left[ x+2\right] = 1$
  Sum rule: differentiate each term separately
3. $\frac{d}{dx}\left[{\left({x}^{2}-1\right)}^{3}\right] = 6 {\left({x}^{2}-1\right)}^{2} x$
  Power rule with the chain rule: d/dx[uⁿ] = n·uⁿ⁻¹·u′
4. $\frac{d}{dx}\left[{x}^{2}-1\right] = 2x$
  Difference rule: differentiate each term separately
3. Steps, part 2
1. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. $f'(x) = 6 {\left({x}^{2}-1\right)}^{2} x \left( x+2\right)+{\left({x}^{2}-1\right)}^{3}$
  Combine and simplify
4. The answer
$f'(x) = 6 {\left({x}^{2}-1\right)}^{2} x \left( x+2\right)+{\left({x}^{2}-1\right)}^{3}$
f'(x) = 6(x² - 1)²x(x + 2) + (x² - 1)³
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: d by d x((x squared minus 1) cubed (x plus 2)). Find the derivative of ((x squared minus 1) cubed (x plus 2)) with respect to x.

Steps, part 1. Product rule: (uv) prime equals u prime v plus uv prime. Sum rule: differentiate each term separately. Power rule with the chain rule: d by d x[u to the power n] equals n times u to the power n minus 1 times u prime. Difference rule: differentiate each term separately.

Steps, part 2. 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 6(x squared minus 1) squaredx(x plus 2) plus (x squared minus 1) cubed. 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 d/dx((x^2 - 1)^3 (x + 2)): Animated Step-by-Step Lesson