If `y=sin^(-1)(x/2)+cos^(-1)(x/2)` then `(dy)/(dx)=`

Options

A. 1
B. -1
C. 0
D. 2

Answer

To solve the problem, we need to differentiate the expression \( y = \sin^{-1} \left( \frac{x}{2} \right) + \cos^{-1} \left( \frac{x}{2} \right) \).

### Step-by-Step Solution:

1. **Write the Expression**:
\[ y = \sin^{-1} \left( \frac{x}{2} \right) + \cos^{-1} \left( \frac{x}{2} \right) \]

2. **Use the Identity**:
We know that:
\[ \sin^{-1}(u) + \cos^{-1}(u) = \frac{\pi}{2} \]
for any \( u \) in the domain. Here, \( u = \frac{x}{2} \).
Thus, we can rewrite \( y \):
\[ y = \frac{\pi}{2} \]

3. **Differentiate Both Sides**:
Now, we differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\pi}{2} \right) \]

4. **Apply the Constant Rule**:
The derivative of a constant is 0:
\[ \frac{dy}{dx} = 0 \]

### Final Answer:
\[ \frac{dy}{dx} = 0 \]