Using the Differential \(dz\) to Approximate the Change in \(z\)

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 \[ \Delta z\approx dz=f_{x}(1,2)\,dx+f_{y}(1,2)\,dy=2(1) (2) ( 0.1) +(1) ( -0.1) =0.4-0.1=0.3 \]