IF `y = (1)/(2) sin x^(2), (dy)/(dx)` will be :

Options

A. `(1)/(2) cos x^(2)`
B. `x cos x^(2)`
C. `(1)/(2) x^(2) cos x^(2)`
D. `sin x`

Answer

To find the derivative of the function \( y = \frac{1}{2} \sin(x^2) \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \frac{1}{2} \sin(x^2) \]

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we will apply the chain rule. The chain rule states that if you have a composite function \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} \]
In our case, let \( t = x^2 \). Then, we can rewrite \( y \) as:
\[ y = \frac{1}{2} \sin(t) \]

### Step 3: Differentiate \( y \) with respect to \( t \)
Now, we differentiate \( y \) with respect to \( t \):
\[ \frac{dy}{dt} = \frac{1}{2} \cos(t) \]

### Step 4: Differentiate \( t \) with respect to \( x \)
Next, we differentiate \( t \) with respect to \( x \):
\[ t = x^2 \implies \frac{dt}{dx} = 2x \]

### Step 5: Apply the chain rule
Now we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = \left(\frac{1}{2} \cos(t)\right) \cdot (2x) \]

### Step 6: Substitute back for \( t \)
Substituting back \( t = x^2 \):
\[ \frac{dy}{dx} = \frac{1}{2} \cos(x^2) \cdot (2x) = x \cos(x^2) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = x \cos(x^2) \]