Calculus Examples

$r = \frac{1}{2 s} - \frac{8}{5 s^{4}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d s} (r) = \frac{d}{d s} (\frac{1}{2 s} - \frac{8}{5 s^{4}})$

Step 2

The derivative of $r$ with respect to $s$ is $r'$ .

$r'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $\frac{1}{2 s} - \frac{8}{5 s^{4}}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{2 s}] + \frac{d}{d s} [- \frac{8}{5 s^{4}}]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Combine $\frac{1}{2}$ and $s^{- 2}$ .

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

Step 3.2.5

Move $s^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.3

Evaluate $\frac{d}{d s} [- \frac{8}{5 s^{4}}]$ .

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Step 3.3.1

Since $- \frac{8}{5}$ is constant with respect to $s$ , the derivative of $- \frac{8}{5 s^{4}}$ with respect to $s$ is $- \frac{8}{5} \frac{d}{d s} [\frac{1}{s^{4}}]$ .

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

Step 3.3.2

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

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

Step 3.3.3

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 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

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Step 3.3.5.1

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

$- \frac{1}{2 s^{2}} - \frac{8}{5} (- s^{4 \cdot - 2} (4 s^{3}))$

Step 3.3.5.2

Multiply $4$ by $- 2$ .

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

Step 3.3.6

Multiply $4$ by $- 1$ .

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

Step 3.3.7

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

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Step 3.3.7.1

Move $s^{3}$ .

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

Step 3.3.7.2

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

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

Step 3.3.7.3

Subtract $8$ from $3$ .

$- \frac{1}{2 s^{2}} - \frac{8}{5} (- 4 s^{- 5})$

Step 3.3.8

Multiply $- 4$ by $- 1$ .

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

Step 3.3.9

Combine $4$ and $\frac{8}{5}$ .

$- \frac{1}{2 s^{2}} + \frac{4 \cdot 8}{5} s^{- 5}$

Step 3.3.10

Multiply $4$ by $8$ .

$- \frac{1}{2 s^{2}} + \frac{32}{5} s^{- 5}$

Step 3.3.11

Combine $\frac{32}{5}$ and $s^{- 5}$ .

$- \frac{1}{2 s^{2}} + \frac{32 s^{- 5}}{5}$

Step 3.3.12

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

$- \frac{1}{2 s^{2}} + \frac{32}{5 s^{5}}$

Step 4

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

$r' = - \frac{1}{2 s^{2}} + \frac{32}{5 s^{5}}$

Step 5

Replace $r'$ with $\frac{d r}{d s}$ .

$\frac{d r}{d s} = - \frac{1}{2 s^{2}} + \frac{32}{5 s^{5}}$