Question

If $y={(\cos x^2)}^2$, then $\frac{dy}{dx}$ is equal to ?

• A. $-4x\sin 2x^2$
• B. $-x\sin x^2$
• C. $-2x\sin 2x^2$
• D. $-x\cos 2x^2$

Answer 1

The correct option is C

$-2x\sin 2x^2$

Explanation for the correct option:

Finding the $\frac{dy}{dx}$:

Given that,

$y={(\cos x^2)}^2$

Differentiate the given function with respect to $x$:

$\begin{array}{rcl}\frac{dy}{dx}& =& 2\cos x^2\times -\sin x^2\times 2x\left[\because \frac{d\left(\cos x^a\right)}{dx}=a\sin x·x^{a-1},\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}\right]\\ & =& -2x2\sin x^2\cos x^2\\ & =& -2x\sin 2x^2\left[\because \sin 2A=2\sin A\cos A\right]\end{array}$

Hence, option (C) is correct.