The differential coefficient of ` sin (cos (x^(2)))` with respect to s is .

Options

A. ` - 2x sin x^(2) cos (cos x ^(2))`
B. `2x sin (x^(2)) cos (x^(2))`
C. `2x sin (x^(2)) cos (x^(2)) cos x `
D. None of the the above

Explanation

To find the differential coefficient of \( y = \sin(\cos(x^2)) \) with respect to \( x \), we will use the chain rule for differentiation. Here are the steps:

### Step 1: Identify the outer and inner functions
Let:
- \( u = \cos(x^2) \) (inner function)
- \( y = \sin(u) \) (outer function)

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

### Step 3: Differentiate the inner function
Now, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(\cos(x^2)) = -\sin(x^2) \cdot \frac{d}{dx}(x^2) = -\sin(x^2) \cdot 2x \]

### Step 4: Apply the chain rule
Now we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \cos(\cos(x^2)) \cdot (-\sin(x^2) \cdot 2x) \]

### Step 5: Simplify the expression
Thus, we can write:
\[ \frac{dy}{dx} = -2x \sin(x^2) \cos(\cos(x^2)) \]

### Final Answer
The differential coefficient of \( \sin(\cos(x^2)) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -2x \sin(x^2) \cos(\cos(x^2)) \]

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