Chapter 4, Problem 14RE

To determine

The derivative of the given function.

Answer to Problem 14RE

The derivative is $e^{-x}(1-x)$ .

Explanation of Solution

Given information: Given function $y=x{e}^{-x}$

Formula used: $\frac{d(x)}{dx}=1\text{and}\frac{d(e^x)}{dx}=e^x$ with chain and product rule.

Calculation:

  $\begin{array}{l}y=x{e}^{-x}\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ \frac{dy}{dx}=x\times \frac{d(e^{-x})}{d(-x)}\times \frac{d(-x)}{dx}+\frac{d(x)}{dx}\times e^{-x}\\ \frac{dy}{dx}=x\times e^{-x}\times -1+1\times e^{-x}\\ \frac{dy}{dx}=e^{-x}(1-x)\end{array}$