The given function is \( \displaystyle y = \cos(\sin x^2) \) and we need to find the derivative of this function with respect to \( x \).
Solution:
To find \( \displaystyle \frac{dy}{dx} \), we will apply the Chain Rule.
\( \begin{align*} &\frac{dy}{dx} = \frac{d \left( \cos(\sin x^2) \right)}{dx} \\ &= \frac{d \left( \cos(\sin x^2) \right)}{d (\sin x^2)} \cdot \frac{d (\sin x^2)}{d (x^2)} \cdot \frac{d (x^2)}{d x} \\ &= -\sin(\sin x^2) \cdot \cos x^2 \cdot 2x \\ &= -2x \cdot \sin(\sin x^2) \cdot \cos x^2 \\ \end{align*} \)Thus, the derivative of the function is:
\( \boxed{ \frac{dy}{dx} = -2x \cdot \sin(\sin x^2) \cdot \cos x^2 } \)Please let me know in the comments if you find any errors in this solution.