Calculus Examples

$r = \frac{1}{5 s^{4}} - \frac{5}{3 s^{2}}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{5 s^{4}} - \frac{5}{3 s^{2}}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{5 s^{4}}] + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$ .

$\frac{d}{d s} [\frac{1}{5 s^{4}}] + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2

Evaluate $\frac{d}{d s} [\frac{1}{5 s^{4}}]$ .

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Step 1.2.1

Since $\frac{1}{5}$ is constant with respect to $s$ , the derivative of $\frac{1}{5 s^{4}}$ with respect to $s$ is $\frac{1}{5} \frac{d}{d s} [\frac{1}{s^{4}}]$ .

$\frac{1}{5} \frac{d}{d s} [\frac{1}{s^{4}}] + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.2

Rewrite $\frac{1}{s^{4}}$ as ${(s^{4})}^{- 1}$ .

$\frac{1}{5} \frac{d}{d s} [{(s^{4})}^{- 1}] + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{- 1}$ and $g (s) = s^{4}$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $s^{4}$ .

$\frac{1}{5} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d s} [s^{4}]) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.3.2

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

$\frac{1}{5} (- {u_{1}}^{- 2} \frac{d}{d s} [s^{4}]) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.3.3

Replace all occurrences of $u_{1}$ with $s^{4}$ .

$\frac{1}{5} (- {(s^{4})}^{- 2} \frac{d}{d s} [s^{4}]) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 4$ .

$\frac{1}{5} (- {(s^{4})}^{- 2} (4 s^{3})) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.5

Multiply the exponents in ${(s^{4})}^{- 2}$ .

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Step 1.2.5.1

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

$\frac{1}{5} (- s^{4 \cdot - 2} (4 s^{3})) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.5.2

Multiply $4$ by $- 2$ .

$\frac{1}{5} (- s^{- 8} (4 s^{3})) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.6

Multiply $4$ by $- 1$ .

$\frac{1}{5} (- 4 s^{- 8} s^{3}) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.7

Multiply $s^{- 8}$ by $s^{3}$ by adding the exponents.

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Step 1.2.7.1

Move $s^{3}$ .

$\frac{1}{5} (- 4 (s^{3} s^{- 8})) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.7.2

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

$\frac{1}{5} (- 4 s^{3 - 8}) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.7.3

Subtract $8$ from $3$ .

$\frac{1}{5} (- 4 s^{- 5}) + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.8

Combine $- 4$ and $\frac{1}{5}$ .

$\frac{- 4}{5} s^{- 5} + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.9

Combine $\frac{- 4}{5}$ and $s^{- 5}$ .

$\frac{- 4 s^{- 5}}{5} + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.10

Move $s^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 4}{5 s^{5}} + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.2.11

Move the negative in front of the fraction.

$- \frac{4}{5 s^{5}} + \frac{d}{d s} [- \frac{5}{3 s^{2}}]$

Step 1.3

Evaluate $\frac{d}{d s} [- \frac{5}{3 s^{2}}]$ .

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Step 1.3.1

Since $- \frac{5}{3}$ is constant with respect to $s$ , the derivative of $- \frac{5}{3 s^{2}}$ with respect to $s$ is $- \frac{5}{3} \frac{d}{d s} [\frac{1}{s^{2}}]$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} \frac{d}{d s} [\frac{1}{s^{2}}]$

Step 1.3.2

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} \frac{d}{d s} [{(s^{2})}^{- 1}]$

Step 1.3.3

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{- 1}$ and $g (s) = s^{2}$ .

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Step 1.3.3.1

To apply the Chain Rule, set $u_{2}$ as $s^{2}$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d s} [s^{2}])$

Step 1.3.3.2

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- {u_{2}}^{- 2} \frac{d}{d s} [s^{2}])$

Step 1.3.3.3

Replace all occurrences of $u_{2}$ with $s^{2}$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- {(s^{2})}^{- 2} \frac{d}{d s} [s^{2}])$

Step 1.3.4

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- {(s^{2})}^{- 2} (2 s))$

Step 1.3.5

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

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Step 1.3.5.1

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- s^{2 \cdot - 2} (2 s))$

Step 1.3.5.2

Multiply $2$ by $- 2$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- s^{- 4} (2 s))$

Step 1.3.6

Multiply $2$ by $- 1$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- 2 s^{- 4} s)$

Step 1.3.7

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- 2 (s^{1} s^{- 4}))$

Step 1.3.8

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

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- 2 s^{1 - 4})$

Step 1.3.9

Subtract $4$ from $1$ .

$- \frac{4}{5 s^{5}} - \frac{5}{3} (- 2 s^{- 3})$

Step 1.3.10

Multiply $- 2$ by $- 1$ .

$- \frac{4}{5 s^{5}} + 2 (\frac{5}{3}) s^{- 3}$

Step 1.3.11

Combine $2$ and $\frac{5}{3}$ .

$- \frac{4}{5 s^{5}} + \frac{2 \cdot 5}{3} s^{- 3}$

Step 1.3.12

Multiply $2$ by $5$ .

$- \frac{4}{5 s^{5}} + \frac{10}{3} s^{- 3}$

Step 1.3.13

Combine $\frac{10}{3}$ and $s^{- 3}$ .

$- \frac{4}{5 s^{5}} + \frac{10 s^{- 3}}{3}$

Step 1.3.14

Move $s^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{5 s^{5}} + \frac{10}{3 s^{3}}$

Step 1.4

Reorder terms.

$f' (s) = \frac{10}{3 s^{3}} - \frac{4}{5 s^{5}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{10}{3 s^{3}} - \frac{4}{5 s^{5}}$ with respect to $s$ is $\frac{d}{d s} [\frac{10}{3 s^{3}}] + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$ .

$\frac{d}{d s} [\frac{10}{3 s^{3}}] + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2

Evaluate $\frac{d}{d s} [\frac{10}{3 s^{3}}]$ .

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Step 2.2.1

Since $\frac{10}{3}$ is constant with respect to $s$ , the derivative of $\frac{10}{3 s^{3}}$ with respect to $s$ is $\frac{10}{3} \frac{d}{d s} [\frac{1}{s^{3}}]$ .

$\frac{10}{3} \frac{d}{d s} [\frac{1}{s^{3}}] + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.2

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

$\frac{10}{3} \frac{d}{d s} [{(s^{3})}^{- 1}] + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{- 1}$ and $g (s) = s^{3}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $s^{3}$ .

$\frac{10}{3} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d s} [s^{3}]) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.3.2

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

$\frac{10}{3} (- {u_{1}}^{- 2} \frac{d}{d s} [s^{3}]) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $s^{3}$ .

$\frac{10}{3} (- {(s^{3})}^{- 2} \frac{d}{d s} [s^{3}]) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.4

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

$\frac{10}{3} (- {(s^{3})}^{- 2} (3 s^{2})) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.5

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

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Step 2.2.5.1

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

$\frac{10}{3} (- s^{3 \cdot - 2} (3 s^{2})) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.5.2

Multiply $3$ by $- 2$ .

$\frac{10}{3} (- s^{- 6} (3 s^{2})) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.6

Multiply $3$ by $- 1$ .

$\frac{10}{3} (- 3 s^{- 6} s^{2}) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.7

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

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Step 2.2.7.1

Move $s^{2}$ .

$\frac{10}{3} (- 3 (s^{2} s^{- 6})) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.7.2

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

$\frac{10}{3} (- 3 s^{2 - 6}) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.7.3

Subtract $6$ from $2$ .

$\frac{10}{3} (- 3 s^{- 4}) + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.8

Combine $- 3$ and $\frac{10}{3}$ .

$\frac{- 3 \cdot 10}{3} s^{- 4} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.9

Multiply $- 3$ by $10$ .

$\frac{- 30}{3} s^{- 4} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.10

Combine $\frac{- 30}{3}$ and $s^{- 4}$ .

$\frac{- 30 s^{- 4}}{3} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.11

Move $s^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 30}{3 s^{4}} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.12

Cancel the common factor of $- 30$ and $3$ .

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Step 2.2.12.1

Factor $3$ out of $- 30$ .

$\frac{3 \cdot - 10}{3 s^{4}} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.12.2

Cancel the common factors.

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Step 2.2.12.2.1

Factor $3$ out of $3 s^{4}$ .

$\frac{3 \cdot - 10}{3 (s^{4})} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.12.2.2

Cancel the common factor.

$\frac{3 \cdot - 10}{3 s^{4}} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.12.2.3

Rewrite the expression.

$\frac{- 10}{s^{4}} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.2.13

Move the negative in front of the fraction.

$- \frac{10}{s^{4}} + \frac{d}{d s} [- \frac{4}{5 s^{5}}]$

Step 2.3

Evaluate $\frac{d}{d s} [- \frac{4}{5 s^{5}}]$ .

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Step 2.3.1

Since $- \frac{4}{5}$ is constant with respect to $s$ , the derivative of $- \frac{4}{5 s^{5}}$ with respect to $s$ is $- \frac{4}{5} \frac{d}{d s} [\frac{1}{s^{5}}]$ .

$- \frac{10}{s^{4}} - \frac{4}{5} \frac{d}{d s} [\frac{1}{s^{5}}]$

Step 2.3.2

Rewrite $\frac{1}{s^{5}}$ as ${(s^{5})}^{- 1}$ .

$- \frac{10}{s^{4}} - \frac{4}{5} \frac{d}{d s} [{(s^{5})}^{- 1}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = s^{- 1}$ and $g (s) = s^{5}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $s^{5}$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d s} [s^{5}])$

Step 2.3.3.2

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

$- \frac{10}{s^{4}} - \frac{4}{5} (- {u_{2}}^{- 2} \frac{d}{d s} [s^{5}])$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $s^{5}$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- {(s^{5})}^{- 2} \frac{d}{d s} [s^{5}])$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d s} [s^{n}]$ is $n s^{n - 1}$ where $n = 5$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- {(s^{5})}^{- 2} (5 s^{4}))$

Step 2.3.5

Multiply the exponents in ${(s^{5})}^{- 2}$ .

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Step 2.3.5.1

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

$- \frac{10}{s^{4}} - \frac{4}{5} (- s^{5 \cdot - 2} (5 s^{4}))$

Step 2.3.5.2

Multiply $5$ by $- 2$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- s^{- 10} (5 s^{4}))$

Step 2.3.6

Multiply $5$ by $- 1$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- 5 s^{- 10} s^{4})$

Step 2.3.7

Multiply $s^{- 10}$ by $s^{4}$ by adding the exponents.

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Step 2.3.7.1

Move $s^{4}$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- 5 (s^{4} s^{- 10}))$

Step 2.3.7.2

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

$- \frac{10}{s^{4}} - \frac{4}{5} (- 5 s^{4 - 10})$

Step 2.3.7.3

Subtract $10$ from $4$ .

$- \frac{10}{s^{4}} - \frac{4}{5} (- 5 s^{- 6})$

Step 2.3.8

Multiply $- 5$ by $- 1$ .

$- \frac{10}{s^{4}} + 5 (\frac{4}{5}) s^{- 6}$

Step 2.3.9

Combine $5$ and $\frac{4}{5}$ .

$- \frac{10}{s^{4}} + \frac{5 \cdot 4}{5} s^{- 6}$

Step 2.3.10

Multiply $5$ by $4$ .

$- \frac{10}{s^{4}} + \frac{20}{5} s^{- 6}$

Step 2.3.11

Combine $\frac{20}{5}$ and $s^{- 6}$ .

$- \frac{10}{s^{4}} + \frac{20 s^{- 6}}{5}$

Step 2.3.12

Move $s^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{10}{s^{4}} + \frac{20}{5 s^{6}}$

Step 2.3.13

Cancel the common factor of $20$ and $5$ .

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Step 2.3.13.1

Factor $5$ out of $20$ .

$- \frac{10}{s^{4}} + \frac{5 \cdot 4}{5 s^{6}}$

Step 2.3.13.2

Cancel the common factors.

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Step 2.3.13.2.1

Factor $5$ out of $5 s^{6}$ .

$- \frac{10}{s^{4}} + \frac{5 \cdot 4}{5 (s^{6})}$

Step 2.3.13.2.2

Cancel the common factor.

$- \frac{10}{s^{4}} + \frac{5 \cdot 4}{5 s^{6}}$

Step 2.3.13.2.3

Rewrite the expression.

$f'' (s) = - \frac{10}{s^{4}} + \frac{4}{s^{6}}$