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DerivativesCalculusCollegeabout 1:33

Derivative of ((x^2+1)^3)/(e^(x^3)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (6(x² + 1)²x - 3(x² + 1)³x²) / e^x³.

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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 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\frac{{\left({x}^{2}+1\right)}^{3}}{{e}^{{x}^{3}}}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\frac{{\left({x}^{2}+1\right)}^{3}}{{e}^{{x}^{3}}}\right] = \frac{6 {\left({x}^{2}+1\right)}^{2} x-3 {\left({x}^{2}+1\right)}^{3} {x}^{2}}{{e}^{{x}^{3}}}$
  Quotient rule: (u/v)′ = (u′v − uv′)/v²
2. $\frac{d}{dx}\left[{e}^{{x}^{3}}\right] = 3{e}^{{x}^{3}} {x}^{2}$
  Exponential rule: d/dx[eᵘ] = eᵘ·u′
3. $\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
4. $\frac{d}{dx}\left[{\left({x}^{2}+1\right)}^{3}\right] = 6 {\left({x}^{2}+1\right)}^{2} x$
  Power rule with the chain rule: d/dx[uⁿ] = n·uⁿ⁻¹·u′
3. Steps, part 2
1. $\frac{d}{dx}\left[{x}^{2}+1\right] = 2x$
  Sum rule: differentiate each term separately
2. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. $f'(x) = \frac{6 {\left({x}^{2}+1\right)}^{2} x-3 {\left({x}^{2}+1\right)}^{3} {x}^{2}}{{e}^{{x}^{3}}}$
  Combine and simplify
4. The answer
$f'(x) = \frac{6 {\left({x}^{2}+1\right)}^{2} x-3 {\left({x}^{2}+1\right)}^{3} {x}^{2}}{{e}^{{x}^{3}}}$
f'(x) = (6(x² + 1)²x - 3(x² + 1)³x²) / e^x³
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of ((x squared plus 1) cubed)/(e to the power x cubed)). Find the derivative of ((x squared plus 1) cubed) / (e to the power x cubed)) with respect to x.

Steps, part 1. Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared. Exponential rule: d by d x[eᵘ] equals eᵘ times u prime. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. 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.

Steps, part 2. Sum rule: differentiate each term separately. 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 plus 1) squaredx minus 3(x squared plus 1) cubedx squared) / e to the power x 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.

Derivative of ((x^2+1)^3)/(e^(x^3)): Derivatives Lesson