Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d s} (r) = \frac{d}{d s} (\frac{1}{3 s^{2}} - \frac{5}{2 s})$

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 $\frac{1}{3 s^{2}} - \frac{5}{2 s}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{3 s^{2}}] + \frac{d}{d s} [- \frac{5}{2 s}]$ .

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 3.2.3.1

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

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

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 $n = - 1$ .

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $2$ by $- 2$ .

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

Step 3.2.6

Multiply $2$ by $- 1$ .

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

Subtract $4$ from $1$ .

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

Step 3.2.10

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

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

Step 3.2.11

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

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

Step 3.2.12

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

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

Step 3.2.13

Move the negative in front of the fraction.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $- 1$ by $- 1$ .

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

Step 3.3.5

Multiply $\frac{5}{2}$ by $1$ .

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

Step 3.3.6

Combine $\frac{5}{2}$ and $s^{- 2}$ .

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

Step 3.3.7

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

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

Step 3.4

Reorder terms.

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

Step 4

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

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

Step 5

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

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