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Derivative of (cos(3x-2))+(sqrt(x^3)): animated lesson

A narrated, step-by-step derivatives video lesson with captions. Final answer: f'(x) = x²·3/2 / sqrt(x³) - 3sin(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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[\cos\left(3x-2\right)+\sqrt{{x}^{3}}\right]$
2. Steps, part 1
1. $\frac{d}{dx}\left[\cos\left(3x-2\right)+\sqrt{{x}^{3}}\right] = \frac{\frac{{x}^{2}\cdot3}{2}}{\sqrt{{x}^{3}}}-3\sin\left(3x-2\right)$
  Sum rule: differentiate each term separately
2. $\frac{d}{dx}\left[\sqrt{{x}^{3}}\right] = \frac{\frac{{x}^{2}\cdot3}{2}}{\sqrt{{x}^{3}}}$
  Chain rule — d/dx[√u] = u′/(2√u)
3. $\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}$
  Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
4. $\frac{d}{dx}\left[\cos\left(3x-2\right)\right] = -\left(3\sin\left(3x-2\right)\right)$
  Chain rule — d/dx[cos u] = −sin u · u′
3. Steps, part 2
1. $\frac{d}{dx}\left[3x-2\right] = 3$
  Difference rule: differentiate each term separately
2. $\frac{d}{dx}\left[3x\right] = 3$
  Constant multiple rule
3. $f'(x) = \frac{\frac{{x}^{2}\cdot3}{2}}{\sqrt{{x}^{3}}}-3\sin\left(3x-2\right)$
  Combine and simplify
4. The answer
$f'(x) = \frac{\frac{{x}^{2}\cdot3}{2}}{\sqrt{{x}^{3}}}-3\sin\left(3x-2\right)$
f'(x) = x²·3/2 / sqrt(x³) - 3sin(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 (cos(3x minus 2)) plus (sqrt(x cubed)). Find the derivative of (cos(3 x minus 2)) plus (sqrt(x cubed)) with respect to x.

Steps, part 1. Sum rule: differentiate each term separately. Chain rule — d by d x[the square root of u] equals u prime /(2the square root of u. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Chain rule — d by d x[cos u] equals negative sin u times u prime.

Steps, part 2. Difference rule: differentiate each term separately. Constant multiple rule. Combine and simplify.

The answer. So the answer is f prime of x equals x squared times 3 over 2 / sqrt(x cubed) minus 3sin(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 (cos(3x-2))+(sqrt(x^3)): Derivatives Lesson