Using the definition, calculate the derivatives of the function. Then

Step 1
The given function is,
${\displaystyle g\left(t\right)=\frac{1}{t^2}}$
To find the derivative of the function, we use the power rule of differentiation,
${\displaystyle \frac{d}{dx}\left(x^n\right)=n{x}^{n-1}}$
Step 2
Applying differentiation on both sides of the function, we get
${\displaystyle g^{\prime }\left(t\right)=\frac{d}{dt}\left(\frac{1}{t^2}\right)}$
${\displaystyle =\frac{d}{dt}\left(t^{-2}\right)}$
${\displaystyle =-2t^{-2-1}}$
${\displaystyle =-2t^{-3}}$
${\displaystyle =-\frac{2}{t^3}}$
Therefore, the derivative of the given function is ${\displaystyle g^{\prime }\left(t\right)=-\frac{2}{t^3}}$
Step 3
Finding the value of the derivative of the given function for the specified value of the variable, we get
${\displaystyle g^{\prime }(-1)=-\frac{2}{{(-1)}^3}}$
${\displaystyle =-\frac{2}{-1}}$
=2
${\displaystyle g^{\prime }\left(2\right)=-\frac{2}{{\left(2\right)}^3}}$
${\displaystyle =-\frac{2}{8}}$
${\displaystyle =-\frac{1}{4}}$
${\displaystyle g^{\prime }\left(\sqrt{3}\right)=-\frac{2}{{\left(\sqrt{3}\right)}^3}}$
${\displaystyle =-\frac{2}{3\sqrt{3}}}$