Chapter 3.3, Problem 10QR

(a)

To determine

To find: The derivative of the given function with respect to $x$ .

Answer to Problem 10QR

The value of the derivative is $\frac{1}{\pi }$ .

Explanation of Solution

Given Data:

The given function is,

  $f\left(x\right)=\frac{x}{\pi }$

Calculation:

Consider the formula to determine the derivative is,

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Then,

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\text{lim}}\frac{\left(\frac{x+h}{\pi }\right)-\frac{x}{\pi }}{h}\\ =\underset{h\to 0}{\text{lim}}\frac{\left(\frac{x+h-x}{h}\right)}{h}\\ =\frac{1}{\pi }\end{array}$

Thus, the value of the derivative is $\frac{1}{\pi }$ .

(b)

To determine

To find: The derivative of the given function with respect to $x$ .

Answer to Problem 10QR

The value of the derivative is $-\frac{\pi }{x^2}$ .

Explanation of Solution

Given Data:

The given function is,

  $f\left(x\right)=\frac{\pi }{x}$

Calculation:

Consider the formula to determine the derivative is,

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Then,

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\underset{h\to 0}{\text{lim}}\frac{\left(\frac{\pi }{x+h}\right)-\frac{\pi }{x}}{h}\\ =\underset{h\to 0}{\text{lim}}\frac{\left(\frac{\pi x-\pi \left(x+h\right)}{\left(x+h\right)x}\right)}{h}\\ =-\frac{\pi }{x^2}\end{array}$

Thus, the value of the derivative is $-\frac{\pi }{x^2}$ .