Solve a problem

DerivativesCalculusCollegeabout 1:34

Derivative of ((2x)^3)/(arctan(x^2+x+1)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = (24x² + 8(-(2x⁴) - x³) / ((x² + x + 1)² + 1) / atan(x² + x + 1)) / atan(x² + x + 1).

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 (0.7 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\frac{{\left(2x\right)}^{3}}{\tan^{-1}\left({x}^{2}+ x+1\right)}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\frac{{\left(2x\right)}^{3}}{\tan^{-1}\left({x}^{2}+ x+1\right)}\right] = \frac{24{x}^{2}+\frac{\frac{8 \left(-\left(2{x}^{4}\right)-{x}^{3}\right)}{{\left({x}^{2}+ x+1\right)}^{2}+1}}{\tan^{-1}\left({x}^{2}+ x+1\right)}}{\tan^{-1}\left({x}^{2}+ x+1\right)}$
  Quotient rule: (u/v)′ = (u′v − uv′)/v²
2. $\frac{d}{dx}\left[\tan^{-1}\left({x}^{2}+ x+1\right)\right] = \frac{2x+1}{{\left({x}^{2}+ x+1\right)}^{2}+1}$
  Chain rule — d/dx[arctan u] = u′/(1 + u²)
3. $\frac{d}{dx}\left[{x}^{2}+ x+1\right] = 2x+1$
  Sum rule: differentiate each term separately
4. $\frac{d}{dx}\left[{x}^{2}+ x\right] = 2x+1$
  Sum rule: differentiate each term separately
3. Steps, part 2
1. $\frac{d}{dx}\left[{x}^{2}\right] = 2x$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
2. $\frac{d}{dx}\left[{\left(2x\right)}^{3}\right] = 24{x}^{2}$
  Power rule with the chain rule: d/dx[uⁿ] = n·uⁿ⁻¹·u′
3. $\frac{d}{dx}\left[2x\right] = 2$
  Constant multiple rule
4. $f'(x) = \frac{24{x}^{2}+\frac{\frac{8 \left(-\left(2{x}^{4}\right)-{x}^{3}\right)}{{\left({x}^{2}+ x+1\right)}^{2}+1}}{\tan^{-1}\left({x}^{2}+ x+1\right)}}{\tan^{-1}\left({x}^{2}+ x+1\right)}$
  Combine and simplify
4. The answer
$f'(x) = \frac{24{x}^{2}+\frac{\frac{8 \left(-\left(2{x}^{4}\right)-{x}^{3}\right)}{{\left({x}^{2}+ x+1\right)}^{2}+1}}{\tan^{-1}\left({x}^{2}+ x+1\right)}}{\tan^{-1}\left({x}^{2}+ x+1\right)}$
f'(x) = (24x² + 8(-(2x⁴) - x³) / ((x² + x + 1)² + 1) / atan(x² + x + 1)) / atan(x² + x + 1)
5. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of ((2x) cubed)/(arctan(x squared plus x plus 1)). Find the derivative of ((2x) cubed) / (atan(x squared plus x 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[arctan u] equals u prime /(1 plus u squared). Sum rule: differentiate each term separately. Sum rule: differentiate each term separately.

Steps, part 2. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Power rule with the chain rule: d by d x[u to the power n] equals n times u to the power n minus 1 times u prime. Constant multiple rule. Combine and simplify.

The answer. So the answer is f prime of x equals (24x squared plus 8(-(2x to the power 4) minus x cubed) / ((x squared plus x plus 1) squared plus 1) / atan(x squared plus x plus 1)) / atan(x squared plus x 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 ((2x)^3)/(arctan(x^2+x+1)): Derivatives Lesson