Chapter 3.3, Problem 45E

To determine

The rule for the differentiation of the expression.

Answer to Problem 45E

The required rule is $-\frac{f^{\prime }\left(x\right)}{{\left(f\left(x\right)\right)}^2}$ .

Explanation of Solution

Given information:

The expression: $\frac{d}{dx}\left(\frac{1}{f\left(x\right)}\right)$

Formula used:

Differentiation formula:

  $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-v^{\prime }u}{v^2}$

Calculation:

The given expression is $\frac{d}{dx}\left(\frac{1}{f\left(x\right)}\right)$ .

Use the differentiation formula of the above expression.

  $\begin{array}{c}\frac{d}{dx}\left(\frac{1}{f\left(x\right)}\right)=\frac{f\left(x\right)\frac{d}{dx}\left(1\right)-\left(1\right)\frac{d}{dx}f\left(x\right)}{{\left(f\left(x\right)\right)}^2}\\ =\frac{f\left(x\right)\left(0\right)-f^{\prime }\left(x\right)}{{\left(f\left(x\right)\right)}^2}\\ =-\frac{f^{\prime }\left(x\right)}{{\left(f\left(x\right)\right)}^2}\end{array}$

Hence, the required rule is $-\frac{f^{\prime }\left(x\right)}{{\left(f\left(x\right)\right)}^2}$ .