Answer
$$f_x(x,y) =-4y ,\ \ f_y(x,y) =-4x+18y^2 , \ \ f_x(2,-1) =4\
f_y(-4,3) =178$$
Work Step by Step
Given $$f(x, y)=-4 x y+6 y^{3}+5$$
Since
\begin{align*}
f_x(x,y)&=\frac{\partial f }{\partial x}\left(-4 x y+6 y^{3}+5\right)\\
&=-4y
\end{align*}
\begin{align*}
f_y(x,y)&=\frac{\partial f }{\partial y}\left(-4 x y+6 y^{3}+5\right)\\
&=-4x+18y^2
\end{align*}
and
\begin{align*}
f_x(2,-1) &=4\\
f_y(-4,3)&=178
\end{align*}
