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Derivative of (arctan(x^2+1))+(arctan(x^2+1)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = 4x / ((x² + 1)² + 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}+1\right)+\tan^{-1}\left({x}^{2}+1\right)\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}+1\right)+\tan^{-1}\left({x}^{2}+1\right)\right] = \frac{4x}{{\left({x}^{2}+1\right)}^{2}+1}$
  Sum rule: differentiate each term separately
2. $\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}+1\right)\right] = \frac{2x}{{\left({x}^{2}+1\right)}^{2}+1}$
  Chain rule — d/dx[arctan u] = u′/(1 + u²)
3. $\frac{d}{dx}\left[{x}^{2}+1\right] = 2x$
  Sum rule: differentiate each term separately
4. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
3. Steps, part 2
1. $f'(x) = \frac{4x}{{\left({x}^{2}+1\right)}^{2}+1}$
  Combine and simplify
4. The answer
$f'(x) = \frac{4x}{{\left({x}^{2}+1\right)}^{2}+1}$
f'(x) = 4x / ((x² + 1)² + 1)
5. 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 plus 1)) plus (arctan(x squared plus 1)). Find the derivative of (atan(x squared plus 1)) plus (atan(x squared plus 1)) with respect to x.

Steps, part 1. Sum rule: differentiate each term separately. Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared). Sum 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.

Steps, part 2. Combine and simplify.

The answer. So the answer is f prime of x equals 4x / ((x squared plus 1) squared 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+1))+(arctan(x^2+1)) (Lesson)