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DerivativesCalculusCollegeabout 1:01

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

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

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 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Finding a derivative
    ddx[2x−1x+3]\frac{d}{dx}\left[\frac{2x-1}{x+3}\right]
  2. 2. Steps, part 1
    1. ddx[2x−1x+3]=1−2xx+3+2x+3\frac{d}{dx}\left[\frac{2x-1}{x+3}\right] = \frac{\frac{1-2x}{x+3}+2}{x+3}
      Quotient rule: (u/v)′ = (u′v − uv′)/v²
    2. ddx[x+3]=1\frac{d}{dx}\left[ x+3\right] = 1
      Sum rule: differentiate each term separately
    3. ddx[2x−1]=2\frac{d}{dx}\left[2x-1\right] = 2
      Difference rule: differentiate each term separately
    4. ddx[2x]=2\frac{d}{dx}\left[2x\right] = 2
      Constant multiple rule
  3. 3. Steps, part 2
    1. f′(x)=1−2xx+3+2x+3f'(x) = \frac{\frac{1-2x}{x+3}+2}{x+3}
      Combine and simplify
  4. 4. The answer
    f′(x)=1−2xx+3+2x+3f'(x) = \frac{\frac{1-2x}{x+3}+2}{x+3}

    f'(x) = ((1 - 2x) / (x + 3) + 2) / (x + 3)

  5. 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((2x minus 1)/(x plus 3)). Find the derivative of ((2 x minus 1) / (x plus 3)) with respect to x.

Steps, part 1. Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared. Sum rule: differentiate each term separately. Difference rule: differentiate each term separately. Constant multiple rule.

Steps, part 2. Combine and simplify.

The answer. So the answer is f prime of x equals ((1 minus 2x) / (x plus 3) plus 2) / (x plus 3). 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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Find d/dx((2x - 1)/(x + 3)): Derivatives Step-by-Step Lesson