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

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