Chapter 3.1, Problem 5E

To determine

To calculate: The derivative of the given function at the given value of a using the definition $f^{\prime }\left(a\right)=\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ .

Answer to Problem 5E

The required value is $-\frac{1}{4}$ .

Explanation of Solution

Given information:

The function: $f\left(x\right)=\frac{1}{x}$ , $a=2$

Calculation:

The given function is $f\left(x\right)=\frac{1}{x}$ .

Write the given definition using the above function as:

  $f^{\prime }\left(a\right)=\underset{x\to a}{\lim }\frac{\frac{1}{x}-\frac{1}{a}}{x-a}$

Substitute 2 for a in the above limit and simplify.

  $\begin{array}{c}{f}^{\prime }\left(2\right)=\underset{x\to 2}{\lim }\left[\frac{\frac{1}{x}-\frac{1}{2}}{x-2}\right]\\ f^{\prime }\left(2\right)=\underset{x\to 2}{\lim }\left[\frac{2-x}{2x}\cdot \frac{1}{x-2}\right]\\ f^{\prime }\left(2\right)=\underset{x\to 2}{\lim }\left[\frac{-\cancel{\left(x-2\right)}}{2x}\cdot \frac{1}{\cancel{\left(x-2\right)}}\right]\\ f^{\prime }\left(2\right)=\underset{x\to 2}{\lim }\left[\frac{-1}{2x}\right]\end{array}$

Substitute the 2 for x in the limit and simplify.

  $\begin{array}{c}{f}^{\prime }\left(2\right)=\frac{-1}{2\cdot \left(2\right)}\\ =-\frac{1}{4}\end{array}$

Hence, the required value is $-\frac{1}{4}$ .