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Derivative of (ln(x^2+1))/(sqrt(3x-2)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (2x / (x² + 1) + ln(x² + 1)·-3/2 / sqrt(3 x - 2)²) / sqrt(3 x - 2).

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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 (0.6 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\frac{\ln\left({x}^{2}+1\right)}{\sqrt{3x-2}}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\frac{\ln\left({x}^{2}+1\right)}{\sqrt{3x-2}}\right] = \frac{\frac{2x}{{x}^{2}+1}+\frac{\frac{\ln\left({x}^{2}+1\right)\cdot-3}{2}}{{\sqrt{3x-2}}^{2}}}{\sqrt{3x-2}}$
  Quotient rule: (u/v)′ = (u′v − uv′)/v²
2. $\frac{d}{dx}\left[\sqrt{3x-2}\right] = \frac{\frac{3}{2}}{\sqrt{3x-2}}$
  Chain rule — d/dx[√u] = u′/(2√u)
3. $\frac{d}{dx}\left[3x-2\right] = 3$
  Difference rule: differentiate each term separately
4. $\frac{d}{dx}\left[3x\right] = 3$
  Constant multiple rule
3. Steps, part 2
1. $\frac{d}{dx}\left[\ln\left({x}^{2}+1\right)\right] = \frac{2x}{{x}^{2}+1}$
  Chain rule — d/dx[ln u] = u′/u
2. $\frac{d}{dx}\left[{x}^{2}+1\right] = 2x$
  Sum rule: differentiate each term separately
3. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
4. $f'(x) = \frac{\frac{2x}{{x}^{2}+1}+\frac{\frac{\ln\left({x}^{2}+1\right)\cdot-3}{2}}{{\sqrt{3x-2}}^{2}}}{\sqrt{3x-2}}$
  Combine and simplify
4. The answer
$f'(x) = \frac{\frac{2x}{{x}^{2}+1}+\frac{\frac{\ln\left({x}^{2}+1\right)\cdot-3}{2}}{{\sqrt{3x-2}}^{2}}}{\sqrt{3x-2}}$
f'(x) = (2x / (x² + 1) + ln(x² + 1)·-3/2 / sqrt(3 x - 2)²) / sqrt(3 x - 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(x squared plus 1))/(sqrt(3x minus 2)). Find the derivative of (ln(x squared plus 1)) / (sqrt(3 x minus 2)) 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[the square root of u] equals u prime /(2the square root of u. Difference rule: differentiate each term separately. Constant multiple rule.

Steps, part 2. Chain rule — d by d x[ln u] equals u prime /u. 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. Combine and simplify.

The answer. So the answer is f prime of x equals (2x / (x squared plus 1) plus ln(x squared plus 1) times negative 3 over 2 / sqrt(3 x minus 2) squared) / sqrt(3 x minus 2). 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(x^2+1))/(sqrt(3x-2)): Derivatives Lesson