Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified.

$g\left(t\right)=\frac{1}{{t}^{2}};{g}^{\prime}(-1),{g}^{\prime}\left(2\right),{g}^{\prime}\left(\sqrt{3}\right)$

tricotasu
2021-09-28
Answered

Using the definition, calculate the derivatives of the function. Then find the values of the derivatives as specified.

$g\left(t\right)=\frac{1}{{t}^{2}};{g}^{\prime}(-1),{g}^{\prime}\left(2\right),{g}^{\prime}\left(\sqrt{3}\right)$

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casincal

Answered 2021-09-29
Author has **82** answers

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

$g}^{\prime}(-1)=-\frac{2}{{(-1)}^{3}$

$=-\frac{2}{-1}$

=2

$g}^{\prime}\left(2\right)=-\frac{2}{{\left(2\right)}^{3}$

$=-\frac{2}{8}$

$=-\frac{1}{4}$

$g}^{\prime}\left(\sqrt{3}\right)=-\frac{2}{{\left(\sqrt{3}\right)}^{3}$

$=-\frac{2}{3\sqrt{3}}$

The given function is,

To find the derivative of the function, we use the power rule of differentiation,

Step 2

Applying differentiation on both sides of the function, we get

Therefore, the derivative of the given function is

Step 3

Finding the value of the derivative of the given function for the specified value of the variable, we get

=2

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