The given function is \(\displaystyle y = \sin(\cos(x^3)) \) and we need to find the derivative of this function with respect to \( x \).
Solution:
We will apply the Chain Rule to find the derivative of this function.
\( \displaystyle y = \sin(\cos(x^3)) \\ \displaystyle \frac{dy}{dx} = \frac{d}{dx} \left[ \sin(\cos(x^3)) \right] \\ \displaystyle = \frac{d}{d\left(\cos(x^3) \right)} \left[ \sin(\cos(x^3)) \right] \cdot \frac{d}{d(x^3)} (\cos(x^3)) \cdot \frac{d}{dx}x^3 \\ \displaystyle = \cos(\cos(x^3)) \cdot (-\sin(x^3)) \cdot 3x^2 \\ \displaystyle = -3x^2 \cdot \sin(x^3) \cdot \cos(\cos(x^3)) \\ \)Thus, the derivative of the function is:
\( \boxed{\frac{dy}{dx} = -3x^2 \cdot \sin(x^3) \cdot \cos(\cos(x^3))} \\ \)Please let me know in the comments if you find any errors in this solution.