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DerivativesCalculusCollegeabout 1:15

Derivative of x^4 - 3x^3 + 2x - 7: animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = 4x³ + 2 - 9x².

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. 1. The problem
    Finding a derivative
    ddx[x4−3x3+2x−7]\frac{d}{dx}\left[{x}^{4}-3{x}^{3}+2x-7\right]
  2. 2. Steps, part 1
    1. ddx[x4−3x3+2x−7]=4x3+2−9x2\frac{d}{dx}\left[{x}^{4}-3{x}^{3}+2x-7\right] = 4{x}^{3}+2-9{x}^{2}
      Difference rule: differentiate each term separately
    2. ddx[x4−3x3+2x]=4x3+2−9x2\frac{d}{dx}\left[{x}^{4}-3{x}^{3}+2x\right] = 4{x}^{3}+2-9{x}^{2}
      Sum rule: differentiate each term separately
    3. ddx[2x]=2\frac{d}{dx}\left[2x\right] = 2
      Constant multiple rule
    4. ddx[x4−3x3]=4x3−9x2\frac{d}{dx}\left[{x}^{4}-3{x}^{3}\right] = 4{x}^{3}-9{x}^{2}
      Difference rule: differentiate each term separately
  3. 3. Steps, part 2
    1. ddx[3x3]=9x2\frac{d}{dx}\left[3{x}^{3}\right] = 9{x}^{2}
      Constant multiple rule
    2. ddx[x3]=3x2\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
    3. ddx[x4]=4x3\frac{d}{dx}\left[{x}^{4}\right] = 4{x}^{3}
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
    4. f′(x)=4x3+2−9x2f'(x) = 4{x}^{3}+2-9{x}^{2}
      Combine and simplify
  4. 4. The answer
    f′(x)=4x3+2−9x2f'(x) = 4{x}^{3}+2-9{x}^{2}

    f'(x) = 4x³ + 2 - 9x²

  5. 5. Check
    • Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of x to the power 4 minus 3x cubed plus 2x minus 7. Find the derivative of x to the power 4 minus 3 x cubed plus 2 x minus 7 with respect to x.

Steps, part 1. Difference rule: differentiate each term separately. Sum rule: differentiate each term separately. Constant multiple rule. Difference rule: differentiate each term separately.

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. 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 4x cubed plus 2 minus 9x 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.

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Derivative of x^4 - 3x^3 + 2x - 7: Derivatives Lesson