constuct and symplify the differnce quotient for x/(2-x)

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.

construct and simplify the difference quotient for x/(2-x)

This page provides a detailed step-by-step solution on how to construct and simplify the difference quotient for the function f(x) = x/(2 - x). This includes using the difference quotient formula and algebraic simplifications. Suitable for students studying calculus.

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