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DerivativesCalculusCollegeabout 0:59

Derivative of sqrt(1 - x^2): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = -(x / sqrt(1 - 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Finding a derivative
    ddx[1−x2]\frac{d}{dx}\left[\sqrt{1-{x}^{2}}\right]
  2. 2. Step by step
    1. ddx[1−x2]=−x1−x2\frac{d}{dx}\left[\sqrt{1-{x}^{2}}\right] = -\frac{x}{\sqrt{1-{x}^{2}}}
      Chain rule — d/dx[√u] = u′/(2√u)
    2. ddx[1−x2]=−(2x)\frac{d}{dx}\left[1-{x}^{2}\right] = -\left(2x\right)
      Difference rule: differentiate each term separately
    3. ddx[x2]=2x\frac{d}{dx}\left[{x}^{2}\right] = 2x
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
    4. f′(x)=−x1−x2f'(x) = -\frac{x}{\sqrt{1-{x}^{2}}}
      Combine and simplify
  3. 3. The answer
    f′(x)=−x1−x2f'(x) = -\frac{x}{\sqrt{1-{x}^{2}}}

    f'(x) = -(x / sqrt(1 - x²))

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

Transcript

The problem. Here is the problem: derivative of sqrt(1 minus x squared). Find the derivative of sqrt(1 minus x squared) with respect to x.

Step by step. Chain rule — d by d x[the square root of u] equals u prime /(2the square root of u. Difference 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 -(x / sqrt(1 minus 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.

More lessons like this

Derivative of sqrt(1 - x^2): Derivatives Step-by-Step Lesson