Question

Find the derivative of \( y = \sin(x^2) \) with respect to \( x \).

• A. \( \cos(x^2) \)
• B. \( 2x \cos(x^2) \)
• C. \( \sin(x^2) \cdot 2x \)
• D. \( \cos(x^2) \cdot x \)

Hint:

Apply the chain rule for composite functions like \( \sin(x^2) \).

Answer 1

The Correct Option is B

Step 1: Apply the chain rule. Set \( u = x^2 \), which makes \( y = \sin(u) \).
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] Step 2: Calculate the derivatives:
\[ \frac{dy}{du} = \cos(u) = \cos(x^2), \quad \frac{du}{dx} = 2x \] \[ \frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x \cos(x^2) \] Step 3: Confirmation: The chain rule confirms that the derivative of the inner function \( x^2 \) is multiplied. This aligns with option (2).