Question

find the difference quotient $\frac{k(x + h)-k(x)}{h}$ for the following function and simplify it by rationalizing the numerator. $k(x)=sqrt{x + 6}$ $\frac{k(x + h)-k(x)}{h}=square$ (simplify your answer. rationalize the numerator.)

Explanation:

Step1: Find $k(x + h)$

Given $k(x)=\sqrt{x + 6}$, then $k(x + h)=\sqrt{(x + h)+6}=\sqrt{x+h + 6}$.

Step2: Substitute into the difference - quotient formula

The difference - quotient is $\frac{k(x + h)-k(x)}{h}=\frac{\sqrt{x + h+6}-\sqrt{x + 6}}{h}$.

Step3: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator $\sqrt{x + h+6}+\sqrt{x + 6}$.
\[

$$\begin{align*} \frac{\sqrt{x + h+6}-\sqrt{x + 6}}{h}&=\frac{(\sqrt{x + h+6}-\sqrt{x + 6})(\sqrt{x + h+6}+\sqrt{x + 6})}{h(\sqrt{x + h+6}+\sqrt{x + 6})}\\ &=\frac{(x + h+6)-(x + 6)}{h(\sqrt{x + h+6}+\sqrt{x + 6})}\\ &=\frac{x+h + 6-x - 6}{h(\sqrt{x + h+6}+\sqrt{x + 6})}\\ &=\frac{h}{h(\sqrt{x + h+6}+\sqrt{x + 6})} \end{align*}$$

\]

Step4: Simplify the expression

Cancel out the common factor $h$ in the numerator and denominator.
We get $\frac{1}{\sqrt{x + h+6}+\sqrt{x + 6}}$.

Answer:

$\frac{1}{\sqrt{x + h+6}+\sqrt{x + 6}}$