Solve a problem

DerivativesCalculusCollegeabout 0:54

Find d/dx(x e^(-x)): animated lesson

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

SpeedLanguage

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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Finding a derivative
$\frac{d}{dx}\left[ x {e}^{- x}\right]$
2. Step by step
1. $\frac{d}{dx}\left[ x {e}^{- x}\right] = \frac{1- x}{{e}^{x}}$
  Product rule: (uv)′ = u′v + uv′
2. $\frac{d}{dx}\left[{e}^{- x}\right] = -\frac{1}{{e}^{x}}$
  Exponential rule: d/dx[eᵘ] = eᵘ·u′
3. $f'(x) = \frac{1- x}{{e}^{x}}$
  Combine and simplify
3. The answer
$f'(x) = \frac{1- x}{{e}^{x}}$
f'(x) = (1 - x) / e^x
4. Check
- Cross-checked numerically against an independent symbolic differentiator at 8 sample points ✓

Transcript

The problem. Here is the problem: d by d x(x e to the power negative x). Find the derivative of (x e to the power negative x) with respect to x.

Step by step. Product rule: (uv) prime equals u prime v plus uv prime. Exponential rule: d by d x[eᵘ] equals eᵘ times u prime. Combine and simplify.

The answer. So the answer is f prime of x equals (1 minus x) / e to the power 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.

Find d/dx(x e^(-x)): Derivatives Step-by-Step Lesson