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Derivative of (sin(x^3))((2x)^3): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = 24(x⁵·cos(x³) + sin(x³)·x²).

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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
$\frac{d}{dx}\left[\sin\left({x}^{3}\right) \left({\left(2x\right)}^{3}\right)\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\sin\left({x}^{3}\right) \left({\left(2x\right)}^{3}\right)\right] = 24 \left({x}^{5} \cos\left({x}^{3}\right)+\sin\left({x}^{3}\right) {x}^{2}\right)$
  Product rule: (uv)′ = u′v + uv′
2. $\frac{d}{dx}\left[{\left(2x\right)}^{3}\right] = 24{x}^{2}$
  Power rule with the chain rule: d/dx[uⁿ] = n·uⁿ⁻¹·u′
3. $\frac{d}{dx}\left[2x\right] = 2$
  Constant multiple rule
4. $\frac{d}{dx}\left[\sin\left({x}^{3}\right)\right] = 3\cos\left({x}^{3}\right) {x}^{2}$
  Chain rule — d/dx[sin u] = cos u · u′
3. Steps, part 2
1. $\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. $f'(x) = 24 \left({x}^{5} \cos\left({x}^{3}\right)+\sin\left({x}^{3}\right) {x}^{2}\right)$
  Combine and simplify
4. The answer
$f'(x) = 24 \left({x}^{5} \cos\left({x}^{3}\right)+\sin\left({x}^{3}\right) {x}^{2}\right)$
f'(x) = 24(x⁵·cos(x³) + sin(x³)·x²)
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of (sin(x cubed))((2x) cubed). Find the derivative of (sin(x cubed)) ((2x) cubed) with respect to x.

Steps, part 1. Product rule: (uv) prime equals u prime v plus uv prime. 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. Constant multiple rule. Chain rule — d by d x[sin u] equals cos u times u prime.

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 24(x to the power 5 times cos(x cubed) plus sin(x cubed) times x 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.

Derivative of (sin(x^3))((2x)^3): Derivatives Lesson