Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
DerivativesCalculusCollegeabout 1:16

Derivative of (e^(3x-2))+(ln(x^3)): animated lesson

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

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. 1. The problem
    Finding a derivative
    ddx[e3x−2+ln⁡(x3)]\frac{d}{dx}\left[{e}^{3x-2}+\ln\left({x}^{3}\right)\right]
  2. 2. Steps, part 1
    1. ddx[e3x−2+ln⁡(x3)]=3(e3x−2+1x)\frac{d}{dx}\left[{e}^{3x-2}+\ln\left({x}^{3}\right)\right] = 3 \left({e}^{3x-2}+\frac{1}{x}\right)
      Sum rule: differentiate each term separately
    2. ddx[ln⁡(x3)]=3x\frac{d}{dx}\left[\ln\left({x}^{3}\right)\right] = \frac{3}{x}
      Chain rule — d/dx[ln u] = u′/u
    3. ddx[x3]=3x2\frac{d}{dx}\left[{x}^{3}\right] = 3{x}^{2}
      Power rule: d/dx[xⁿ] = n·xⁿ⁻¹
    4. ddx[e3x−2]=3e3x−2\frac{d}{dx}\left[{e}^{3x-2}\right] = 3{e}^{3x-2}
      Exponential rule: d/dx[eᵘ] = eᵘ·u′
  3. 3. Steps, part 2
    1. ddx[3x−2]=3\frac{d}{dx}\left[3x-2\right] = 3
      Difference rule: differentiate each term separately
    2. ddx[3x]=3\frac{d}{dx}\left[3x\right] = 3
      Constant multiple rule
    3. f′(x)=3(e3x−2+1x)f'(x) = 3 \left({e}^{3x-2}+\frac{1}{x}\right)
      Combine and simplify
  4. 4. The answer
    f′(x)=3(e3x−2+1x)f'(x) = 3 \left({e}^{3x-2}+\frac{1}{x}\right)

    f'(x) = 3(e^(3 x - 2) + 1 / x)

  5. 5. Check
    • Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: derivative of (e to the power 3x minus 2)) plus (ln(x cubed)). Find the derivative of (e to the power 3 x minus 2)) plus (ln(x cubed)) with respect to x.

Steps, part 1. Sum rule: differentiate each term separately. Chain rule — d by d x[ln u] equals u prime /u. Power rule: d by d x[x to the power n] equals n times x to the power n minus 1. Exponential rule: d by d x[eᵘ] equals eᵘ 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 3(e to the power 3 x minus 2) plus 1 / x). 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.

More lessons like this

Derivative of (e^(3x-2))+(ln(x^3)): Derivatives Lesson