Answer
$dz=\frac{7}{225}$
Work Step by Step
We are given $z=f(x,y)=\frac{y^{2}+3x}{y^{2}-x}$
$f_{x}(x,y)=\frac{3(y^{2}-x)-(-1)(y^{2}+3x)}{(y^{2}-x)^{2}}$
$f_{x}(x,y)=\frac{3y^{2}-3x+y^{2}+3x}{(y^{2}-x)^{2}}$
$f_{x}(x,y)=\frac{4y^{2}}{(y^{2}-x)^{2}}$
$f_{y}(x,y)=\frac{2y(y^{2}-x)-2y(y^{2}+3x)}{(y^{2}-x)^{2}}$
$f_{}(x,y)=\frac{2y^{3}-2xy-2y^{3}-6xy}{(y^{2}-x)^{2}}$
$f_{}(x,y)=\frac{-8xy}{(y^{2}-x)^{2}}$
By the definition $dz=f_{x}(x,y).dx+f_{y}(x,y)dy$
$dz=(\frac{4y^{2}}{(y^{2}-x)^{2}})dx+(\frac{-8xy}{(y^{2}-x)^{2}})dy$
With $x=4, y=-4, dx=0.01, dy=0.03$
$dz=\frac{7}{225}$
