Calculus Examples

$s = t^{2} + \frac{9}{t^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (s) = \frac{d}{d t} (t^{2} + \frac{9}{t^{2}})$

Step 2

The derivative of $s$ with respect to $t$ is $s'$ .

$s'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Evaluate $\frac{d}{d t} [\frac{9}{t^{2}}]$ .

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Step 3.2.1

Since $9$ is constant with respect to $t$ , the derivative of $\frac{9}{t^{2}}$ with respect to $t$ is $9 \frac{d}{d t} [\frac{1}{t^{2}}]$ .

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

Step 3.2.2

Rewrite $\frac{1}{t^{2}}$ as ${(t^{2})}^{- 1}$ .

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

Step 3.2.3

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 $g (t) = t^{2}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $t^{2}$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u$ with $t^{2}$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $2$ by $- 2$ .

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

Step 3.2.6

Multiply $2$ by $- 1$ .

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

Step 3.2.7

Raise $t$ to the power of $1$ .

$2 t + 9 (- 2 (t^{1} t^{- 4}))$

Step 3.2.8

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

$2 t + 9 (- 2 t^{1 - 4})$

Step 3.2.9

Subtract $4$ from $1$ .

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

Step 3.2.10

Multiply $- 2$ by $9$ .

$2 t - 18 t^{- 3}$

Step 3.3

Simplify.

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Step 3.3.1

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

$2 t - 18 \frac{1}{t^{3}}$

Step 3.3.2

Combine terms.

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Step 3.3.2.1

Combine $- 18$ and $\frac{1}{t^{3}}$ .

$2 t + \frac{- 18}{t^{3}}$

Step 3.3.2.2

Move the negative in front of the fraction.

$2 t - \frac{18}{t^{3}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$s' = 2 t - \frac{18}{t^{3}}$

Step 5

Replace $s'$ with $\frac{d s}{d t}$ .

$\frac{d s}{d t} = 2 t - \frac{18}{t^{3}}$