College Algebra, Chapter 3, 3.1, Section 3.1, Problem 38

Given the function $\displaystyle f(x) = \frac{1}{x+1}$. Find $f(a)$, $f(a+h)$ and the difference quotient $\displaystyle \frac{f(a+h) - f(a)}{h}$ where $h \neq 0$

For $f(a)$
$\displaystyle f(a) = \frac{1}{a+1}$ Replace $x$ by $a$

For $f(a+h)$
$\displaystyle f(a+h) = \frac{1}{a+h+1}$ Replace $x$ by $(a+h)$

For $\displaystyle \frac{f(a+h)-f(a)}{h}$

$
\begin{equation}
\begin{aligned}
\frac{f(a-h)-f(a)}{h} &= \frac{\frac{1}{a+h+1} - \frac{1}{a+h} }{h} && \text{Substitute } f(a+h) = \frac{1}{a+h+1} \text{ and } f(a) = \frac{1}{a+h}\\
\\
&= \frac{a+h-(a+h+1)}{h(a+h)(a+h+1)} && \text{Get the LCD}\\
\\
&= \frac{a+h-a-h-1}{h(a+h)(a+h+1)} && \text{Simplify}\\
\\
&= \frac{-1}{h(a+h)(a+h+1)}
\end{aligned}
\end{equation}
$

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