Question

Find the derivative of the function given by \(f(x) = \sin(x^2)\)

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

Hint:

Do not confuse \(\sin(x^2)\) with \(\sin^2(x) = (\sin x)^2\), whose derivative is \(2\sin x\cos x = \sin(2x)\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The derivative of a composite function \(f(g(x))\) is calculated using the chain rule of differential calculus.
Key Formula or Approach:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]
\[ \frac{d}{du}(\sin u) = \cos u, \quad \frac{d}{dx}(x^2) = 2x \]

Step 2: Detailed Explanation:
Let \(u = g(x) = x^2\), so that \(f(x) = \sin(u)\).
Differentiating the outer function with respect to \(u\):
\[ \frac{df}{du} = \cos(u) = \cos(x^2) \]
Differentiating the inner function with respect to \(x\):
\[ \frac{du}{dx} = 2x \]
Applying the chain rule:
\[ f'(x) = \frac{df}{du} \cdot \frac{du}{dx} = \cos(x^2) \cdot (2x) = 2x \cos(x^2) \]

Step 3: Final Answer:
Therefore, the derivative is \(2 \, x \cos x^2\), corresponding to option (D).