Differentiation of `sin(x^2)` w.r.t. x is

Options

A. `cos(x^2)`
B. `2xcos(x^2)`
C. `x^2cos(x^2)`
D. `-cos(2x)`

Explanation

To differentiate \( \sin(x^2) \) with respect to \( x \), 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 function \( y = \sin(x^2) \), we can identify:
- The outer function \( y = \sin(u) \) where \( u = x^2 \)
- The inner function \( u = x^2 \)

### Step 2: Differentiate the outer function
We first differentiate the outer function \( y = \sin(u) \) with respect to \( u \):
\[ \frac{dy}{du} = \cos(u) \]

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

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

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

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