Chapter 4.4, Problem 17E

To determine

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

Answer to Problem 17E

The derivative is $\frac{dy}{dx}=\frac{-1}{x}$ .

Explanation of Solution

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

  $y=\ln \left(\frac{1}{x}\right)$

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

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left(\ln \left(\frac{1}{x}\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{1}{\left(\frac{1}{x}\right)}\times \frac{d\left(\frac{1}{x}\right)}{dx}\\ \frac{dy}{dx}=x\times \frac{-1}{x^2}\\ \frac{dy}{dx}=\frac{-1}{x}\end{array}$

So the derivative is $\frac{dy}{dx}=\frac{-1}{x}$ .