`x/(sqrt(1-x^2))`

Answer

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

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

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

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

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find the derivatives of \( u \) and \( v \):
- \( \frac{du}{dx} = 1 \)
- To differentiate \( v \), we use the chain rule:
\[ v = (1 - x^2)^{1/2} \]
Using the chain rule:
\[ \frac{dv}{dx} = \frac{1}{2}(1 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}} \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:
\[ \frac{d}{dx}\left(\frac{x}{\sqrt{1 - x^2}}\right) = \frac{\sqrt{1 - x^2} \cdot 1 - x \cdot \left(\frac{-x}{\sqrt{1 - x^2}}\right)}{(1 - x^2)} \]

### Step 4: Simplify the Expression
Now, simplify the expression:
\[ = \frac{\sqrt{1 - x^2} + \frac{x^2}{\sqrt{1 - x^2}}}{1 - x^2} \]
Combine the terms in the numerator:
\[ = \frac{\frac{(1 - x^2) + x^2}{\sqrt{1 - x^2}}}{1 - x^2} = \frac{\frac{1}{\sqrt{1 - x^2}}}{1 - x^2} \]

### Step 5: Final Expression
Thus, the derivative simplifies to:
\[ \frac{1}{(1 - x^2)^{3/2}} \]

### Final Answer
The differential coefficient of the function \( f(x) = \frac{x}{\sqrt{1 - x^2}} \) is:
\[ f'(x) = \frac{1}{(1 - x^2)^{3/2}} \]