The given function is \( \displaystyle y = \sin \sqrt{\sin \sqrt{x}} \) and we need to find the derivative of this function with respect to \( x \).
Solution:
To find \( \frac{dy}{dx} \), we will apply the Chain Rule repeatedly since the function is a composition of multiple functions.
\( \displaystyle \frac{dy}{dx} = \frac{d[\sin \sqrt{\sin \sqrt{x}}]}{d(\sqrt{\sin \sqrt{x}})} \cdot \frac{d(\sqrt{\sin \sqrt{x}})}{d(\sin \sqrt{x})} \cdot \frac{d(\sin \sqrt{x})}{d(\sqrt{x})} \cdot \frac{d(\sqrt x)}{dx} \\ \displaystyle = \cos \sqrt{\sin \sqrt{x}} \cdot \frac{1}{2 \sqrt{\sin \sqrt{x}}} \cdot \cos \sqrt{x} \cdot \frac{1}{2 \sqrt{x}} \\ \displaystyle = \cos \sqrt{\sin \sqrt{x}} \cdot \frac{\cos \sqrt{x}}{2 \sqrt{\sin \sqrt{x}}} \cdot \frac{1}{2 \sqrt{x}} \\ \)Thus, the derivative of the function is:
\( \boxed{ \frac{dy}{dx} = \frac{\cos \sqrt{\sin \sqrt{x}} \cdot \cos \sqrt{x}}{4 \sqrt{x} \cdot \sqrt{\sin \sqrt{x}}} } \\ \)Please let me know in the comments if you find any errors in this solution.