Calculus Examples

$\frac{1}{3 s^{2}} - \frac{6}{5 s^{4}}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{3 s^{2}} - \frac{6}{5 s^{4}}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{3 s^{2}}] + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$ .

$\frac{d}{d s} [\frac{1}{3 s^{2}}] + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2

Evaluate $\frac{d}{d s} [\frac{1}{3 s^{2}}]$ .

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Step 2.1

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

$\frac{1}{3} \frac{d}{d s} [\frac{1}{s^{2}}] + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.2

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

$\frac{1}{3} \frac{d}{d s} [{(s^{2})}^{- 1}] + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 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 $g (s) = s^{2}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $s^{2}$ .

$\frac{1}{3} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d s} [s^{2}]) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 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 $n = - 1$ .

$\frac{1}{3} (- {u_{1}}^{- 2} \frac{d}{d s} [s^{2}]) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $s^{2}$ .

$\frac{1}{3} (- {(s^{2})}^{- 2} \frac{d}{d s} [s^{2}]) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.4

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

$\frac{1}{3} (- {(s^{2})}^{- 2} (2 s)) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{3} (- s^{2 \cdot - 2} (2 s)) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.5.2

Multiply $2$ by $- 2$ .

$\frac{1}{3} (- s^{- 4} (2 s)) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.6

Multiply $2$ by $- 1$ .

$\frac{1}{3} (- 2 s^{- 4} s) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.7

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

$\frac{1}{3} (- 2 (s^{1} s^{- 4})) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.8

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

$\frac{1}{3} (- 2 s^{1 - 4}) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.9

Subtract $4$ from $1$ .

$\frac{1}{3} (- 2 s^{- 3}) + \frac{d}{d s} [- \frac{6}{5 s^{4}}]$

Step 2.10

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 2.11

Combine $\frac{- 2}{3}$ and $s^{- 3}$ .

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

Step 2.12

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

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

Step 2.13

Move the negative in front of the fraction.

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

Step 3

Evaluate $\frac{d}{d s} [- \frac{6}{5 s^{4}}]$ .

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Step 3.1

Since $- \frac{6}{5}$ is constant with respect to $s$ , the derivative of $- \frac{6}{5 s^{4}}$ with respect to $s$ is $- \frac{6}{5} \frac{d}{d s} [\frac{1}{s^{4}}]$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} \frac{d}{d s} [\frac{1}{s^{4}}]$

Step 3.2

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

$- \frac{2}{3 s^{3}} - \frac{6}{5} \frac{d}{d s} [{(s^{4})}^{- 1}]$

Step 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 $g (s) = s^{4}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{2}$ as $s^{4}$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d s} [s^{4}])$

Step 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 $n = - 1$ .

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

Step 3.3.3

Replace all occurrences of $u_{2}$ with $s^{4}$ .

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

Step 3.4

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

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- {(s^{4})}^{- 2} (4 s^{3}))$

Step 3.5

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

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Step 3.5.1

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

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- s^{4 \cdot - 2} (4 s^{3}))$

Step 3.5.2

Multiply $4$ by $- 2$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- s^{- 8} (4 s^{3}))$

Step 3.6

Multiply $4$ by $- 1$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- 4 s^{- 8} s^{3})$

Step 3.7

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

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Step 3.7.1

Move $s^{3}$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- 4 (s^{3} s^{- 8}))$

Step 3.7.2

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

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- 4 s^{3 - 8})$

Step 3.7.3

Subtract $8$ from $3$ .

$- \frac{2}{3 s^{3}} - \frac{6}{5} (- 4 s^{- 5})$

Step 3.8

Multiply $- 4$ by $- 1$ .

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

Step 3.9

Combine $4$ and $\frac{6}{5}$ .

$- \frac{2}{3 s^{3}} + \frac{4 \cdot 6}{5} s^{- 5}$

Step 3.10

Multiply $4$ by $6$ .

$- \frac{2}{3 s^{3}} + \frac{24}{5} s^{- 5}$

Step 3.11

Combine $\frac{24}{5}$ and $s^{- 5}$ .

$- \frac{2}{3 s^{3}} + \frac{24 s^{- 5}}{5}$

Step 3.12

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

$- \frac{2}{3 s^{3}} + \frac{24}{5 s^{5}}$