For the function $z=f(x, y)=x^{2}y-1,$ use the differential $dz$ to approximate the change in $z$ from $(1,2)$ to $(1.1,1.9)$.

Solution Example 2 shows $f$ is differentiable and $f_{x}(x,y) =2xy$ and $f_{y}(x,y) =x^{2}.$

Let $( x_{0},y_{0}) =( 1,2)$ and $( x_{0}+\Delta x,y_{0}+\Delta y) =( 1.1,1.9) .$ Then $$dx=\Delta x=1.1-1=0.1, \hbox{ and } dy=\Delta y=1.9-2= -0.1.$$

Using (4), an approximation to the change in $z$ is $$\Delta z\approx dz=f_{x}(1,2)\,dx+f_{y}(1,2)\,dy=2(1) (2) ( 0.1) +(1) ( -0.1) =0.4-0.1=0.3$$