Answer
$$ z_u= \cos (u^2v) (2uv) = 2uv \cos (u^2v), \quad \\
z_v= \cos (u^2v) (u^2) = u^2\cos (u^2v).
$$
Work Step by Step
Recall that $(\sin x)'=\cos x$.
Recall that $(x^n)'=nx^{n-1}$
Since $ z=\sin (u^2v)$, then we have
$$ z_u= \cos (u^2v) (2uv) = 2uv \cos (u^2v), \quad\\
z_v= \cos (u^2v) (u^2) = u^2\cos (u^2v).
$$
