Chapter 1.3, Problem 54E

Below are the steps in the simplification of the difference quotient for $f\left(x\right)=\sqrt{x}$

(see Example 8).

Provide a brief justification for each step.

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

a. a) $=\frac{\sqrt{x+h}-\sqrt{x}}{h}\cdot \left(\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}\right)$

Multiplying by 1

b. b) $\frac{x+h+\sqrt{x}\sqrt{x+h}-\sqrt{x}\sqrt{x+h}-x}{h\left(\sqrt{x+h}+\sqrt{x}\right)}$

Expanding (multiplying) the numerator

c. c) $\frac{x+h-x}{h\left(\sqrt{x+h}+\sqrt{x}\right)}\sqrt{x}\sqrt{x+h}-\sqrt{x}\sqrt{x+h}=0$

d. d) $=\frac{h}{h\left(\sqrt{x+h}+\sqrt{x}\right)}x-x=0$

e. e) $=\frac{1}{\sqrt{x+h}+\sqrt{x}}$

Assuming $h\ne 0,\frac{h}{h}=1$