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Derivative of (ln(sin(x)))+(arctan(2x)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = cos(x) / sin(x) + 2 / (4x² + 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
$\frac{d}{dx}\left[\ln\left(\sin\left( x\right)\right)+\tan^{-1}\left(2x\right)\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\ln\left(\sin\left( x\right)\right)+\tan^{-1}\left(2x\right)\right] = \frac{\cos\left( x\right)}{\sin\left( x\right)}+\frac{2}{4{x}^{2}+1}$
  Sum rule: differentiate each term separately
2. $\frac{d}{dx}\left[\tan^{-1}\left(2x\right)\right] = \frac{2}{4{x}^{2}+1}$
  Chain rule — d/dx[arctan u] = u′/(1 + u²)
3. $\frac{d}{dx}\left[2x\right] = 2$
  Constant multiple rule
4. $\frac{d}{dx}\left[\ln\left(\sin\left( x\right)\right)\right] = \frac{\cos\left( x\right)}{\sin\left( x\right)}$
  Chain rule — d/dx[ln u] = u′/u
3. Steps, part 2
1. $\frac{d}{dx}\left[\sin\left( x\right)\right] = \cos\left( x\right)$
  d/dx[sin x] = cos x
2. $f'(x) = \frac{\cos\left( x\right)}{\sin\left( x\right)}+\frac{2}{4{x}^{2}+1}$
  Combine and simplify
4. The answer
$f'(x) = \frac{\cos\left( x\right)}{\sin\left( x\right)}+\frac{2}{4{x}^{2}+1}$
f'(x) = cos(x) / sin(x) + 2 / (4x² + 1)
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of (ln(sin(x))) plus (arctan(2x)). Find the derivative of (ln(sin(x))) plus (atan(2 x)) 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). Constant multiple rule. Chain rule — d by d x[ln u] equals u prime /u.

Steps, part 2. d by d x[sin x] equals cos x. Combine and simplify.

The answer. So the answer is f prime of x equals cos(x) / sin(x) plus 2 / (4x 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 (ln(sin(x)))+(arctan(2x)): Derivatives Lesson