Chapter 4.4, Problem 8E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 8E

The derivative is $\frac{dy}{dx}=x^2{e}^x+x{e}^x-e^x$ .

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y=x^2{e}^x-x{e}^x$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left(x^2{e}^x-x{e}^x\right)}{dx}\\ \frac{dy}{dx}=x^2\times \frac{d\left(e^x\right)}{dx}+e^x\times \frac{d\left(x^2\right)}{dx}-x\times \frac{d\left(e^x\right)}{dx}-e^x\times \frac{dx}{dx}\\ \frac{dy}{dx}=x^2\times e^x+2x\times e^x-x\times e^x-e^x\\ \frac{dy}{dx}=x^2{e}^x+x{e}^x-e^x\end{array}$

So the derivative is $\frac{dy}{dx}=x^2{e}^x+x{e}^x-e^x$ .