Find the differentiation of the function:
f ( x ) = `x/ (1 − x^ 2)` at x=2

Options

A. `5/7`
B. `2/3`
C. `5/9`
D. none of the above

Answer

To find the differentiation of the function \( f(x) = \frac{x}{1 - x^2} \) at \( x = 2 \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( f'(x) \) is given by:

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

where \( u = x \) and \( v = 1 - x^2 \).

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

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

### Step 3: Apply the Quotient Rule
Now we apply the quotient rule:
\[ f'(x) = \frac{(1 - x^2)(1) - (x)(-2x)}{(1 - x^2)^2} \]

### Step 4: Simplify the Expression
Now, simplify the numerator:
\[ f'(x) = \frac{1 - x^2 + 2x^2}{(1 - x^2)^2} = \frac{1 + x^2}{(1 - x^2)^2} \]

### Step 5: Evaluate at \( x = 2 \)
Now we need to evaluate \( f'(x) \) at \( x = 2 \):
\[ f'(2) = \frac{1 + (2)^2}{(1 - (2)^2)^2} = \frac{1 + 4}{(1 - 4)^2} = \frac{5}{(-3)^2} = \frac{5}{9} \]

### Final Answer
Thus, the differentiation of the function \( f(x) \) at \( x = 2 \) is:
\[ f'(2) = \frac{5}{9} \]

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