Calculus Examples

$g (t) = t^{2} - \frac{9}{t^{3}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $t^{2} - \frac{9}{t^{3}}$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- \frac{9}{t^{3}}]$ .

$\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- \frac{9}{t^{3}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$2 t + \frac{d}{d t} [- \frac{9}{t^{3}}]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{9}{t^{3}}]$ .

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Step 2.1

Since $- 9$ is constant with respect to $t$ , the derivative of $- \frac{9}{t^{3}}$ with respect to $t$ is $- 9 \frac{d}{d t} [\frac{1}{t^{3}}]$ .

$2 t - 9 \frac{d}{d t} [\frac{1}{t^{3}}]$

Step 2.2

Rewrite $\frac{1}{t^{3}}$ as ${(t^{3})}^{- 1}$ .

$2 t - 9 \frac{d}{d t} [{(t^{3})}^{- 1}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{3}$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $t^{3}$ .

$2 t - 9 (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{3}])$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 1$ .

$2 t - 9 (- u^{- 2} \frac{d}{d t} [t^{3}])$

Step 2.3.3

Replace all occurrences of $u$ with $t^{3}$ .

$2 t - 9 (- {(t^{3})}^{- 2} \frac{d}{d t} [t^{3}])$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 3$ .

$2 t - 9 (- {(t^{3})}^{- 2} (3 t^{2}))$

Step 2.5

Multiply the exponents in ${(t^{3})}^{- 2}$ .

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Step 2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 t - 9 (- t^{3 \cdot - 2} (3 t^{2}))$

Step 2.5.2

Multiply $3$ by $- 2$ .

$2 t - 9 (- t^{- 6} (3 t^{2}))$

Step 2.6

Multiply $3$ by $- 1$ .

$2 t - 9 (- 3 t^{- 6} t^{2})$

Step 2.7

Multiply $t^{- 6}$ by $t^{2}$ by adding the exponents.

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Step 2.7.1

Move $t^{2}$ .

$2 t - 9 (- 3 (t^{2} t^{- 6}))$

Step 2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 t - 9 (- 3 t^{2 - 6})$

Step 2.7.3

Subtract $6$ from $2$ .

$2 t - 9 (- 3 t^{- 4})$

Step 2.8

Multiply $- 3$ by $- 9$ .

$2 t + 27 t^{- 4}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 t + 27 \frac{1}{t^{4}}$

Step 3.2

Combine $27$ and $\frac{1}{t^{4}}$ .

$2 t + \frac{27}{t^{4}}$