Differentiate the function with respect to $x$: $\cos(x^{3}) \cdot \sin^{2}(x^{5})$

(N/A) Let $y = \cos(x^{3}) \cdot \sin^{2}(x^{5})$.
Using the product rule $\frac{d}{dx}[u \cdot v] = u \frac{dv}{dx} + v \frac{du}{dx}$:
$\frac{dy}{dx} = \cos(x^{3}) \cdot \frac{d}{dx}[\sin^{2}(x^{5})] + \sin^{2}(x^{5}) \cdot \frac{d}{dx}[\cos(x^{3})]$
Applying the chain rule:
$\frac{dy}{dx} = \cos(x^{3}) \cdot [2 \sin(x^{5}) \cdot \cos(x^{5}) \cdot 5x^{4}] + \sin^{2}(x^{5}) \cdot [-\sin(x^{3}) \cdot 3x^{2}]$
Simplifying the expression:
$\frac{dy}{dx} = 10x^{4} \sin(x^{5}) \cos(x^{5}) \cos(x^{3}) - 3x^{2} \sin(x^{3}) \sin^{2}(x^{5})$