Answer
$g_u=10uv(u^2v-v^3)^4$, $g_v=5(u^2-3v^2)(u^2v-v^3)^4$.
Work Step by Step
$g(u,v)=(u^2v-v^3)^5$
In order to find $g_u$ we treat $v$ as a constant and differentiate with respect to $u$.
$g_u=10uv(u^2v-v^3)^4$
In order to find $g_v$ we treat $u$ as a constant and differentiate with respect to $v$.
$g_v=5(u^2-3v^2)(u^2v-v^3)^4$
