Answer
$\frac{∂f}{∂x}=-0.2x^{-0.9}y^{0.9}$
$\frac{∂f}{∂y}=-1.8x^{0.1}y^{-0.1}$
$\frac{∂f}{∂x}|_{(1,-1)}$ and $\frac{∂f}{∂y}|_{(1,-1)}$ are not defined.
Work Step by Step
$f(x,y)=-2x^{0.1}y^{0.9}$
$\frac{∂f}{∂x}=-2(0.1)x^{-0.9}y^{0.9}=-0.2x^{-0.9}y^{0.9}$
$\frac{∂f}{∂y}=-2x^{0.1}(0.9)y^{-0.1}=-1.8x^{0.1}y^{-0.1}$
$\frac{∂f}{∂x}|_{(1,-1)}$ and $\frac{∂f}{∂y}|_{(1,-1)}$ are not defined because $(1,-1)$ is not in the domain of $f(x,y)=-2x^{0.1}y^{0.9}$
$f(1,-1)=-2(1)^{0.1}(-1)^{0.9}$
But $(-1)^{0.9}$ has no solution.
