Differentiating $f(x,y) = 5x^{3}y^{2}$ with respect to $y$ gives $10x^{3}y$, with the $x^{3}$ carried through untouched. Why does the $x$ term survive unchanged?

D. Because a partial derivative holds $x$ fixed, so $5x^{3}$ is constant.

Holding the other variable fixed turns its whole factor into a coefficient. This is exactly why the marginal product of labour still depends on the capital stock.