Question

\( \frac{d}{dx} \left( \sin x^2 \right) = 2x \cos x^2 \)

Hint:

When differentiating composite functions, apply the chain rule: \( \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} \).

Answer 1

Step 1: Applying the chain rule. 
We are given the function \( y = \sin(x^2) \). To differentiate this, we use the chain rule, which states: \[ \frac{d}{dx} \sin(u) = \cos(u) \cdot \frac{du}{dx} \] where \( u = x^2 \). 
Step 2: Differentiating the inner function. 
The derivative of \( u = x^2 \) is \( \frac{du}{dx} = 2x \). 
Step 3: Final derivative. 
Applying the chain rule, we get: \[ \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot 2x \] which simplifies to: \[ 2x \cos(x^2) \] Conclusion: 
Thus, the statement is True.