The derivative of sin^2(x^2) with respect to x is

The correct answer is Option (4) → \(2x\sin(2x^{2})\)

The function is a square of a sine of \(x^{2}\), so the chain rule is used through three layers:

    \( \displaystyle \frac{d}{dx}\sin^{2}(x^{2}) = 2\sin(x^{2}) \cdot \cos(x^{2}) \cdot 2x \)

Using \(2\sin\theta\cos\theta = \sin 2\theta\) with \(\theta = x^{2}\):

    \( = 2x \cdot 2\sin(x^{2})\cos(x^{2}) \)

    \( = 2x\sin(2x^{2}) \)

Leaving out the last factor 2x, the derivative of the innermost \(x^{2}\), is the usual slip.