Answer
$\dfrac{1}{2}$ and $\dfrac{3}{4}$
Work Step by Step
We know that $f_x(x_0,y_0)=\lim\limits_{h \to 0} \dfrac{f(x_0+h,y_0)-f(x_0,y_0)}{h}$
$f_x(-2,3)=\lim\limits_{h \to 0} \dfrac{f(-2+h,3)-f(-2,3)}{h}=\lim\limits_{h \to 0} \dfrac{\sqrt {2h+4}-2}{h}=\lim\limits_{h \to 0} \dfrac{2}{\sqrt {2h+4}-2}=\dfrac{1}{2}$
Also, $f_y(x_0,y_0)=\lim\limits_{h \to 0} \dfrac{f(x_0,y_0+h)-f(x_0,y_0)}{h}$
Now,$f_y(-2,3)=\lim\limits_{h \to 0} \dfrac{f(-2,3+h)-f(-2,3)}{h}=\lim\limits_{h \to 0} \dfrac{\sqrt {3h+4}-2}{h}=\lim\limits_{h \to 0} \dfrac{3}{\sqrt {3h+4}-2}=\dfrac{3}{4}$
