EXAMPLE $4$

$($a$)$ If

z $=$ f$($x$,$ y$)=$ x$2+6$xy $$ y$2,$

find the differential dz$.$

$($b$)$ If x changes from $2$ to $2\mathrm{.}06$ and y changes from $3$ to $2\mathrm{.}93,$ compare the values of $$z and dz$.$

SOLUTION

$($a$)$ The definition of the differential gives

dz$=$

$$z$$x

dx $+$

$$z$$y

dy

$=$

dx $+$

dy$.$

$($b$)$ Putting

x $=2,$ dx $=$x $=0\mathrm{.}06,$ y $=3,$ and dy $=$y $=0\mathrm{.}07,$

we get

dz$=$

$(2)+6(3)$

$(0\mathrm{.}06)+$

$6(2)2(3)$

$$

$=.$

The increment of z is as follows. $($Round your final answer to two decimal places.$)$

$$z$=$f$(2\mathrm{.}06,2\mathrm{.}93)$ f$(2,3)$

$=$

$2$

$+6(2\mathrm{.}06)(2\mathrm{.}93)(2\mathrm{.}93)2$

$$

$22+6(2)(3)2$

$$

$=.$

Notice that

$$z $$ dz

is easier to compute.