Answer
a. $\frac{\partial g}{\partial x}=12y$ or $\frac{\partial g}{\partial x}=12x$;
b. $\frac{\partial g}{\partial y}=4$ or $\frac{\partial g}{\partial y}=12x$;
c. 54;
d. 32.
Work Step by Step
$g_{x}(x,y) =8+12xy$
$g_{y}(x,y)=6x^{2}+4y$
a. $\frac{\partial g}{\partial x}=g_{xx}(x,y)=12y$
$\frac{\partial g}{\partial x}=g_{xy}(x,y)=12x$
b. $\frac{\partial g}{\partial y}=g_{yy}(x,y)=4$
$\frac{\partial g}{\partial y}=g_{yx}(x,y)=12x$
c. Since $\frac{\partial z}{\partial x}=6x^{2} + 4y$
$\frac{\partial z}{\partial x}(-3,0)=6(-3)^{2}+4(0)=54$
d. $g_{x}(2,1)=8+12(2)(1)=32$
