If f(x) = `(x-2)/(x+2) x ne -2` is then f'(x) equal to?

Options

A. `(4)/(x+2)^2`
B. `(x+2)/(4(x-2))`
C. `(x+2)/(x-2)`
D. `(2(x+1))/(1-x)`

Answer

To find the derivative of the function \( f(x) = \frac{x-2}{x+2} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form of \( \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = \frac{v \cdot u' - u \cdot v'}{v^2} \]

where \( u = x - 2 \) and \( v = x + 2 \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x - 2 \)
- \( v = x + 2 \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find the derivatives \( u' \) and \( v' \):
- \( u' = \frac{d}{dx}(x - 2) = 1 \)
- \( v' = \frac{d}{dx}(x + 2) = 1 \)

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ f'(x) = \frac{(x + 2)(1) - (x - 2)(1)}{(x + 2)^2} \]

### Step 4: Simplify the Expression
Now simplify the numerator:

\[ f'(x) = \frac{(x + 2) - (x - 2)}{(x + 2)^2} \]

Distributing the negative sign in the second term:

\[ f'(x) = \frac{x + 2 - x + 2}{(x + 2)^2} \]

This simplifies to:

\[ f'(x) = \frac{4}{(x + 2)^2} \]

### Final Result
Thus, the derivative \( f'(x) \) is:

\[ f'(x) = \frac{4}{(x + 2)^2} \]