Answer
a) $15x^{2}+15hx+5h^{2}$
b) $6y+3h$
c) $15x^{2}$
d) $6y$
Work Step by Step
a) $\frac{f(x+h,y)-f(x,y)}{h}=\frac{5(x+h)^{3}+3y^{2}-(5x^{3}+3y^{2})}{h}=15x^{2}+15hx+5h^{2}$
b) $\frac{f(x,y+h)-f(x,y)}{h}=\frac{5x^{3}+3(y+h)^{2}-(5x^{3}+3y^{2})}{h}=6y+3h$
c) $\lim\limits_{h \to 0}\frac{f(x+h,y)-f(x,y)}{h}=\lim\limits_{h \to 0}(15x^{2}+15hx+5h^{2})=15x^{2}+15(0)x+5(0)^{2}=15x^{2}$
d) $\lim\limits_{h \to 0}\frac{f(x,y+h)-f(x,y)}{h}=\lim\limits_{h \to 0}(6y+3h)=6y+3(0)=6y$
