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

### Step 1: Identify the outer and inner functions
In the expression \(y = \frac{1}{2} \sin(x^2)\), we can identify:
- The outer function: \(u = \sin(v)\) where \(v = x^2\)
- The inner function: \(v = x^2\)

### Step 2: Differentiate the outer function
The derivative of the outer function \(u = \sin(v)\) with respect to \(v\) is:
\[ \frac{du}{dv} = \cos(v) \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \(v = x^2\) with respect to \(x\):
\[ \frac{dv}{dx} = 2x \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(v) \cdot 2x \]

### Step 5: Substitute back the inner function
Now we substitute \(v = x^2\) back into the equation:
\[ \frac{dy}{dx} = \cos(x^2) \cdot 2x \]

### Step 6: Include the constant factor
Since we have a constant factor of \(\frac{1}{2}\) in the original function, we multiply the result by \(\frac{1}{2}\):
\[ \frac{dy}{dx} = \frac{1}{2} \cdot 2x \cos(x^2) = x \cos(x^2) \]

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

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