Differentiate the functions with respect to x.

The given function is $\cos x^{3} \cdot \sin ^{2}\left(x^{5}\right)$.

$\frac{d}{d x}\left[\cos x^{3} \cdot \sin ^{2}\left(x^{5}\right)\right]=\sin ^{2}\left(x^{5}\right) \times \frac{d}{d x}\left(\cos x^{3}\right)+\cos x^{3} \times \frac{d}{d x}\left[\sin ^{2}\left(x^{5}\right)\right]$\

$=\sin ^{2}\left(x^{5}\right) \times\left(-\sin x^{3}\right) \times \frac{d}{d x}\left(x^{3}\right)+\cos x^{3} \times 2 \sin \left(x^{5}\right) \cdot \frac{d}{d x}\left[\sin x^{5}\right]$

$=-\sin x^{3} \sin ^{2}\left(x^{5}\right) \times 3 x^{2}+2 \sin x^{5} \cos x^{3} \cdot \cos x^{5} \times \frac{d}{d x}\left(x^{5}\right)$

$=-3 x^{2} \sin x^{3} \cdot \sin ^{2}\left(x^{5}\right)+2 \sin x^{5} \cos x^{5} \cos x^{3} \cdot \times 5 x^{4}$

$=10 x^{4} \sin x^{5} \cos x^{5} \cos x^{3}-3 x^{2} \sin x^{3} \sin ^{2}\left(x^{5}\right)$