Resolved: "Using the definition, calculate the derivatives of the..."

nitraiddQ

Answered question

2021-05-26

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

g (t) = \frac{1}{t^{2}} . g^{'} (- 1), g^{'} (2), g^{'} (\sqrt{3})

Macsen Nixon

Skilled2021-05-27Added 117 answers

Step 1
The given function is,

g (t) = \frac{1}{t^{2}}

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

\frac{d}{d x} (x^{n}) = n x^{n - 1}

Step 2
Applying differentiation on both sides of the function, we get

g^{'} (t) = \frac{d}{d t} (\frac{1}{t^{2}})

= \frac{d}{d t} (t^{- 2})

= - 2 t^{- 2 - 1}

= - 2 t^{- 3}

= - \frac{2}{t^{3}}

Therefore, the derivative of the given function is

g^{'} (t) = - \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^{'} (- 1) = - \frac{2}{{(- 1)}^{3}}

= - \frac{2}{- 1}

g^{'} (2) = - \frac{2}{{(2)}^{3}}

= - \frac{2}{8}

= - \frac{1}{4}

g^{'} (\sqrt{3}) = - \frac{2}{{(\sqrt{3})}^{3}}

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

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