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Derivative of arctan(x^2): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = 2x / (x⁴ + 1).

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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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}\right)\right]$
2. Step by step
1. $\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}\right)\right] = \frac{2x}{{x}^{4}+1}$
  Chain rule — d/dx[arctan u] = u′/(1 + u²)
2. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. $f'(x) = \frac{2x}{{x}^{4}+1}$
  Combine and simplify
3. The answer
$f'(x) = \frac{2x}{{x}^{4}+1}$
f'(x) = 2x / (x⁴ + 1)
4. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of arctan(x squared). Find the derivative of atan(x squared) with respect to x.

Step by step. Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared). 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 / (x to the power 4 plus 1). 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.

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