To construct and simplify the difference quotient for the function $f(x) = \frac{x}{2 - x}$, follow these steps:

The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}$

Step 1: Find $f(x+h)$

Substitute $x+h$ into the function $f(x)$:

$f(x+h) = \frac{x+h}{2 - (x+h)} = \frac{x+h}{2 - x - h}$

Step 2: Set up the difference quotient

Now plug $f(x+h)$ and $f(x)$ into the difference quotient formula:

$\frac{f(x+h) - f(x)}{h} = \frac{\frac{x+h}{2 - x - h} - \frac{x}{2 - x}}{h}$

Step 3: Combine the two fractions in the numerator

We need to combine the fractions:

$\frac{x+h}{2 - x - h} - \frac{x}{2 - x}$

To combine these fractions, find the common denominator:

$\text{Common denominator} = (2 - x - h)(2 - x)$

So the expression becomes:

$\frac{(x+h)(2-x) - x(2 - x - h)}{(2 - x - h)(2 - x)}$

Step 4: Expand the terms in the numerator

Expand the terms in the numerator:

$(x+h)(2 - x) = 2x + 2h - x^2 - xh$

$x(2 - x - h) = 2x - x^2 - xh$

Now subtract the second expression from the first:

$(2x + 2h - x^2 - xh) - (2x - x^2 - xh) = 2h$

Step 5: Simplify the difference quotient

So the difference quotient now becomes:

$\frac{2h}{h(2 - x - h)(2 - x)}$

Cancel out the $h$ in the numerator and denominator:

$\frac{2}{(2 - x - h)(2 - x)}$

This is the simplified form of the difference quotient.

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Do you need further clarification or more details on any part of this solution?

Here are 5 related questions to consider:

1. What happens to the difference quotient as $h$ approaches 0?
2. How does the difference quotient relate to the derivative of the function?
3. Can you simplify the difference quotient for a different function, like $f(x) = \frac{1}{x}$?
4. How does the chain rule apply when differentiating functions like $f(x) = \frac{x}{2 - x}$?
5. Can you find the second derivative of the given function?

Tip: When simplifying algebraic fractions, always look for opportunities to cancel common factors between the numerator and the denominator.