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DerivativesCalculusCollegeabout 1:21

Evaluate second derivative of x^3 - 2x^2: animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f''(x) = 6x - 4.

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

Transcript

The problem. Here is the problem: x cubed minus 2x squared. Find the second derivative of x cubed minus 2 x squared with respect to x.

Steps, part 1. Derivative number 1. Difference rule: differentiate each term separately. Constant multiple rule. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

Steps, part 2. 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. Difference rule: differentiate each term separately.

Steps, part 3. Constant multiple rule. 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.

The answer. So the answer is f prime prime (x) equals 6x minus 4. 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.

Evaluate second derivative of x^3 - 2x^2: Derivatives Lesson