Chapter 13, Problem 23RE

To determine

To determine the derivative of the function at the given number

Answer to Problem 23RE

The derivative of the function at the given number is $\frac{1}{4}$

Explanation of Solution

Given information:

Consider the given information,

  $f\left(x\right)=\frac{x}{x+1}\text{ at }1$

Formula:

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

CALCULATION:

The derivative of the function is,

  $\begin{array}{l}f\left(4\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}\\ =\underset{h\to 0}{\lim }\frac{\frac{a+h}{a+h+1}-\frac{a}{a+1}}{h}\\ =\underset{h\to 0}{\lim }\frac{a^2+ha+a+h-a^2-ah-a}{h\left(a+h+1\right)\left(a+1\right)}\\ =\underset{h\to 0}{\lim }\frac{h}{h\left(a+h+1\right)\left(a+1\right)}\\ =\underset{h\to 0}{\lim }\frac{1}{h\left(a+h+1\right)\left(a+1\right)}\\ =\frac{1}{\left(a+0+1\right)\left(a+1\right)}=\frac{1}{{\left(a+1\right)}^2}\\ f\left(1\right)=\frac{1}{{\left(1+1\right)}^2}=\frac{1}{4}\end{array}$

Therefore, the derivative of the function at the given number is $\frac{1}{4}$ .