Chapter 2.1, Problem 82AYU

To determine

To find the difference quotient $\frac{f\left(x+h\right)-f\left(x\right)}{h}$ .

Answer to Problem 82AYU

  $\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{1}{\sqrt{x+h+1}+\sqrt{x+1}}$

Explanation of Solution

Given:

A function is given as

  $f\left(x\right)=\sqrt{x+1}$

Calculation:

Difference quotient can be calculated as shown below:

  $\begin{array}{l}\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}\\ =\frac{\sqrt{x+h+1}-\sqrt{x+1}}{h}\times \frac{\sqrt{x+h+1}+\sqrt{x+1}}{\sqrt{x+h+1}+\sqrt{x+1}}\\ =\frac{x+h+1-x-1}{h\left(\sqrt{x+h+1}+\sqrt{x+1}\right)}\\ \\ =\frac{h}{h\left(\sqrt{x+h+1}+\sqrt{x+1}\right)}\\ =\frac{1}{\sqrt{x+h+1}+\sqrt{x+1}}\end{array}$

Therefore, the difference quotient is $\frac{1}{\sqrt{x+h+1}+\sqrt{x+1}}$ .