Consider the function $f(x)=x^{-2}$ and express the relationship between a small chan in $x$ and the corresponding change in $y$ in the form $dy=f^{\text{'}}(x)dx.$ Also, find the valu of the differential $dy$ if $x$ changes from $2$ to $1\mathrm{.}9.$

$[$A$]dy=-\frac{1}{x}dx$; value of differential $dy$ is ${?}^{}-\frac{1}{20}.$

$[$B$]dy=\frac{1}{x^2}dx$; value of differential $dy$ is $-\frac{1}{4}.$

$[$C$]dy=-\frac{2}{x^3}dx$; value of differential $dy$ is $\frac{1}{40}.$

$[$D$]dy=-\frac{2}{x^3}dx$; value of differential $dy$ is $-\frac{1}{40}.$

$[$E$]dy=-\frac{1}{x}dx$; value of differential $dy$ is $\frac{1}{20}.$