Solve a problem

DerivativesCalculusCollegeabout 1:20

Derivative of (ln(2x))/(sin(x^2+1)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (sin(x² + 1) / x - 2x·ln(2x)·cos(x² + 1)) / sin(x² + 1)².

SpeedLanguage

Chapters

Transcript

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\frac{\ln\left(2x\right)}{\sin\left({x}^{2}+1\right)}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\frac{\ln\left(2x\right)}{\sin\left({x}^{2}+1\right)}\right] = \frac{\frac{\sin\left({x}^{2}+1\right)}{x}-2x \ln\left(2x\right) \cos\left({x}^{2}+1\right)}{{\sin\left({x}^{2}+1\right)}^{2}}$
  Quotient rule: (u/v)′ = (u′v − uv′)/v²
2. $\frac{d}{dx}\left[\sin\left({x}^{2}+1\right)\right] = 2\cos\left({x}^{2}+1\right) x$
  Chain rule — d/dx[sin u] = cos u · u′
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. $\frac{d}{dx}\left[\ln\left(2x\right)\right] = \frac{1}{x}$
  Chain rule — d/dx[ln u] = u′/u
2. $\frac{d}{dx}\left[2x\right] = 2$
  Constant multiple rule
3. $f'(x) = \frac{\frac{\sin\left({x}^{2}+1\right)}{x}-2x \ln\left(2x\right) \cos\left({x}^{2}+1\right)}{{\sin\left({x}^{2}+1\right)}^{2}}$
  Combine and simplify
4. The answer
$f'(x) = \frac{\frac{\sin\left({x}^{2}+1\right)}{x}-2x \ln\left(2x\right) \cos\left({x}^{2}+1\right)}{{\sin\left({x}^{2}+1\right)}^{2}}$
f'(x) = (sin(x² + 1) / x - 2x·ln(2x)·cos(x² + 1)) / sin(x² + 1)²
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of (ln(2x))/(sin(x squared plus 1)). Find the derivative of (ln(2x)) / (sin(x squared plus 1)) with respect to x.

Steps, part 1. Quotient rule: (u/v) prime equals (u prime v minus uv prime )/v squared. Chain rule — d by d x[sin u] equals cos u times u prime. 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. Chain rule — d by d x[ln u] equals u prime /u. Constant multiple rule. Combine and simplify.

The answer. So the answer is f prime of x equals (sin(x squared plus 1) / x minus 2x times ln(2x) times cos(x squared plus 1)) / sin(x squared plus 1) 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.

Derivative of (ln(2x))/(sin(x^2+1)): Derivatives Lesson