1. ## Difference Quotients

I'm trying to find the Difference Quotient of

F(x) = X / X + 1

So far I've gotten to

F(a) = a / a + 1

F(a + h) = a + h / a + h + 1

2. I'm guessing the function is $f(x) = \frac{x}{x+1}$.

Then the difference quotient is given by

$\frac{f(x+h)-f(x)}{h} = \frac{\frac{x+h}{(x+h)+1}-\frac{x}{x+1}}{h} =\frac{\frac{x+h}{(x+h)+1}\frac{x+1}{x+1}-\frac{x}{x+1}\frac{(x+h)+1}{(x+h)+1}}{h}$
$=\frac{\frac{x^2+x+hx+h}{x^2+x+hx+h+x+1}-\frac{x^2+hx+x}{x^2+hx+x+x+h+1}}{h} =\frac{\frac{x^2+x+hx+h-x^2-hx-x}{x^2+2x+hx+h+1}}{h}$
$=\frac{h}{h(x^2+2x+hx+h+1)} = \frac{1}{x^2+2x+hx+h+1} = \boxed{\frac{1}{(x+1)^2+hx+h}}$.

3. Hello, Jonathan!

Find the Difference Quotient of: . $f(x) \:=\:\frac{x}{x+1}$

So far I've gotten to:

$f(a) \:= \:\frac{a}{a + 1}$

$f(a + h) \:= \:\frac{a + h}{a + h + 1}$ . . . . Good!

We want: . $\frac{f(a+h) - f(a)}{h}$

The numerator is: . $f(a+h) - f(a) \;=\;\frac{a+h}{a+h+1} - \frac{a}{a+1}$

Get a common denominator and subtract:

. . $\frac{a+h}{a+h+1}\cdot\frac{a+1}{a+1} + \frac{a}{a+1}\cdot\frac{a+h+1}{a+h+1} \;=\;\frac{(a+h)(a+1) - (a(a+h+1)}{(a+h+1)(a+1)}$

. . $= \;\frac{a^2 + a +ah + h - a^2 - ah - a}{(a+h+1)(a+1)} \;=\;\frac{h}{(a+h+1)(a+1)}$

Then divide by $h\!:\;\;\frac{f(a+h) - f(a)}{h} \;=\;\frac{1}{h}\cdot\frac{h}{(a+h+1)(a+1)} \;=\;\boxed{\frac{1}{(a+h+1)(a+1)}}$