Chapter 4.4, Problem 20E

To determine

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

Answer to Problem 20E

The derivative is $\frac{dy}{dx}=\ln x$ .

Explanation of Solution

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

  $y=x\ln x-x$

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

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left(x\ln x-x\right)}{dx}\\ \frac{dy}{dx}=x\times \frac{d\left(\ln x\right)}{dx}+\ln x\times \frac{dx}{dx}-\frac{dx}{dx}\\ \frac{dy}{dx}=\frac{x}{x}+\ln x-1\\ \frac{dy}{dx}=\ln x\end{array}$

So the derivative is $\frac{dy}{dx}=\ln x$ .