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DerivativesCalculusCollegeabout 0:56

Find d/dx(e^(-x^2)): animated lesson

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

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 (0.3 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Finding a derivative
    ddx[e−x2]\frac{d}{dx}\left[{e}^{-{x}^{2}}\right]
  2. 2. Step by step
    1. ddx[e−x2]=−2xex2\frac{d}{dx}\left[{e}^{-{x}^{2}}\right] = -\frac{2x}{{e}^{{x}^{2}}}
      Exponential rule: d/dx[eᵘ] = eᵘ·u′
    2. ddx[x2]=2x\frac{d}{dx}\left[{x}^{2}\right] = 2x
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
    3. f′(x)=−2xex2f'(x) = -\frac{2x}{{e}^{{x}^{2}}}
      Combine and simplify
  3. 3. The answer
    f′(x)=−2xex2f'(x) = -\frac{2x}{{e}^{{x}^{2}}}

    f'(x) = -(2x / e^x²)

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

Transcript

The problem. Here is the problem: d by d x(e to the power negative x squared)). Find the derivative of (e to the power negative x squared)) with respect to x.

Step by step. 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. Combine and simplify.

The answer. So the answer is f prime of x equals -(2x / e to the power 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.

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Find d/dx(e^(-x^2)): Derivatives Step-by-Step Lesson