Derivative of cos(sin(x^2)) with respect to x is :

The correct answer is Option (3) → \(-2x\cos(x^2)\sin(\sin(x^2))\)

Let \(y = \cos(\sin(x^2))\). By the chain rule, differentiate from the outside in:

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

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

    \( \displaystyle = -\sin(\sin(x^2))\cdot\cos(x^2)\cdot 2x \)

    \(= -2x\cos(x^2)\sin(\sin(x^2))\)