Calculus Examples

$\frac{1}{2 s} - \frac{7}{3 s^{2}}$

Step 1

By the Sum Rule, the derivative of $\frac{1}{2 s} - \frac{7}{3 s^{2}}$ with respect to $s$ is $\frac{d}{d s} [\frac{1}{2 s}] + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$ .

$\frac{d}{d s} [\frac{1}{2 s}] + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 2

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

Tap for more steps...

Step 2.1

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

$\frac{1}{2} \frac{d}{d s} [\frac{1}{s}] + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 2.2

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

$\frac{1}{2} \frac{d}{d s} [s^{- 1}] + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 2.3

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

$\frac{1}{2} (- s^{- 2}) + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 2.4

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

$- \frac{s^{- 2}}{2} + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 2.5

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

$- \frac{1}{2 s^{2}} + \frac{d}{d s} [- \frac{7}{3 s^{2}}]$

Step 3

Evaluate $\frac{d}{d s} [- \frac{7}{3 s^{2}}]$ .

Tap for more steps...

Step 3.1

Since $- \frac{7}{3}$ is constant with respect to $s$ , the derivative of $- \frac{7}{3 s^{2}}$ with respect to $s$ is $- \frac{7}{3} \frac{d}{d s} [\frac{1}{s^{2}}]$ .

$- \frac{1}{2 s^{2}} - \frac{7}{3} \frac{d}{d s} [\frac{1}{s^{2}}]$

Step 3.2

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} \frac{d}{d s} [{(s^{2})}^{- 1}]$

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

Tap for more steps...

Step 3.3.1

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d s} [s^{2}])$

Step 3.3.2

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- u^{- 2} \frac{d}{d s} [s^{2}])$

Step 3.3.3

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- {(s^{2})}^{- 2} \frac{d}{d s} [s^{2}])$

Step 3.4

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- {(s^{2})}^{- 2} (2 s))$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- s^{2 \cdot - 2} (2 s))$

Step 3.5.2

Multiply $2$ by $- 2$ .

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- s^{- 4} (2 s))$

Step 3.6

Multiply $2$ by $- 1$ .

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- 2 s^{- 4} s)$

Step 3.7

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- 2 (s^{1} s^{- 4}))$

Step 3.8

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

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- 2 s^{1 - 4})$

Step 3.9

Subtract $4$ from $1$ .

$- \frac{1}{2 s^{2}} - \frac{7}{3} (- 2 s^{- 3})$

Step 3.10

Multiply $- 2$ by $- 1$ .

$- \frac{1}{2 s^{2}} + 2 (\frac{7}{3}) s^{- 3}$

Step 3.11

Combine $2$ and $\frac{7}{3}$ .

$- \frac{1}{2 s^{2}} + \frac{2 \cdot 7}{3} s^{- 3}$

Step 3.12

Multiply $2$ by $7$ .

$- \frac{1}{2 s^{2}} + \frac{14}{3} s^{- 3}$

Step 3.13

Combine $\frac{14}{3}$ and $s^{- 3}$ .

$- \frac{1}{2 s^{2}} + \frac{14 s^{- 3}}{3}$

Step 3.14

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

$- \frac{1}{2 s^{2}} + \frac{14}{3 s^{3}}$