Calculus Examples

$r = \frac{1}{4 s^{3}} - \frac{8}{3 s^{2}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d s} (r) = \frac{d}{d s} (\frac{1}{4 s^{3}} - \frac{8}{3 s^{2}})$

Step 2

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

$r'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $\frac{1}{4 s^{3}} - \frac{8}{3 s^{2}}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{4 s^{3}}] + \frac{d}{d s} [- \frac{8}{3 s^{2}}]$ .

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

Step 3.2

Evaluate $\frac{d}{d s} [\frac{1}{4 s^{3}}]$ .

Tap for more steps...

Step 3.2.1

Since $\frac{1}{4}$ is constant with respect to $s$ , the derivative of $\frac{1}{4 s^{3}}$ with respect to $s$ is $\frac{1}{4} \frac{d}{d s} [\frac{1}{s^{3}}]$ .

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

Step 3.2.2

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

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

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 $g (s) = s^{3}$ .

Tap for more steps...

Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

Tap for more steps...

Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$\frac{1}{4} (- s^{- 6} (3 s^{2})) + \frac{d}{d s} [- \frac{8}{3 s^{2}}]$

Step 3.2.6

Multiply $3$ by $- 1$ .

$\frac{1}{4} (- 3 s^{- 6} s^{2}) + \frac{d}{d s} [- \frac{8}{3 s^{2}}]$

Step 3.2.7

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

Tap for more steps...

Step 3.2.7.1

Move $s^{2}$ .

$\frac{1}{4} (- 3 (s^{2} s^{- 6})) + \frac{d}{d s} [- \frac{8}{3 s^{2}}]$

Step 3.2.7.2

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

$\frac{1}{4} (- 3 s^{2 - 6}) + \frac{d}{d s} [- \frac{8}{3 s^{2}}]$

Step 3.2.7.3

Subtract $6$ from $2$ .

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

Step 3.2.8

Combine $- 3$ and $\frac{1}{4}$ .

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

Step 3.2.9

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

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

Step 3.2.10

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

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

Step 3.2.11

Move the negative in front of the fraction.

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

Step 3.3

Evaluate $\frac{d}{d s} [- \frac{8}{3 s^{2}}]$ .

Tap for more steps...

Step 3.3.1

Since $- \frac{8}{3}$ is constant with respect to $s$ , the derivative of $- \frac{8}{3 s^{2}}$ with respect to $s$ is $- \frac{8}{3} \frac{d}{d s} [\frac{1}{s^{2}}]$ .

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

Step 3.3.2

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

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

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

Tap for more steps...

Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

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

Tap for more steps...

Step 3.3.5.1

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

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

Step 3.3.5.2

Multiply $2$ by $- 2$ .

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

Step 3.3.6

Multiply $2$ by $- 1$ .

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

Step 3.3.7

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

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

Step 3.3.8

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

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

Step 3.3.9

Subtract $4$ from $1$ .

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

Step 3.3.10

Multiply $- 2$ by $- 1$ .

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

Step 3.3.11

Combine $2$ and $\frac{8}{3}$ .

$- \frac{3}{4 s^{4}} + \frac{2 \cdot 8}{3} s^{- 3}$

Step 3.3.12

Multiply $2$ by $8$ .

$- \frac{3}{4 s^{4}} + \frac{16}{3} s^{- 3}$

Step 3.3.13

Combine $\frac{16}{3}$ and $s^{- 3}$ .

$- \frac{3}{4 s^{4}} + \frac{16 s^{- 3}}{3}$

Step 3.3.14

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

$- \frac{3}{4 s^{4}} + \frac{16}{3 s^{3}}$

Step 3.4

Reorder terms.

$\frac{16}{3 s^{3}} - \frac{3}{4 s^{4}}$

Step 4

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

$r' = \frac{16}{3 s^{3}} - \frac{3}{4 s^{4}}$

Step 5

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

$\frac{d r}{d s} = \frac{16}{3 s^{3}} - \frac{3}{4 s^{4}}$