Calculus Examples

$\frac{5}{9 w^{4}} + 5 \sqrt[3]{w}$

Step 1

By the Sum Rule, the derivative of $\frac{5}{9 w^{4}} + 5 \sqrt[3]{w}$ with respect to $w$ is $\frac{d}{d w} [\frac{5}{9 w^{4}}] + \frac{d}{d w} [5 \sqrt[3]{w}]$ .

$\frac{d}{d w} [\frac{5}{9 w^{4}}] + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2

Evaluate $\frac{d}{d w} [\frac{5}{9 w^{4}}]$ .

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Step 2.1

Since $\frac{5}{9}$ is constant with respect to $w$ , the derivative of $\frac{5}{9 w^{4}}$ with respect to $w$ is $\frac{5}{9} \frac{d}{d w} [\frac{1}{w^{4}}]$ .

$\frac{5}{9} \frac{d}{d w} [\frac{1}{w^{4}}] + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.2

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

$\frac{5}{9} \frac{d}{d w} [{(w^{4})}^{- 1}] + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = w^{- 1}$ and $g (w) = w^{4}$ .

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Step 2.3.1

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

$\frac{5}{9} (\frac{d}{d u} [u^{- 1}] \frac{d}{d w} [w^{4}]) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.3.2

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

$\frac{5}{9} (- u^{- 2} \frac{d}{d w} [w^{4}]) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.3.3

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

$\frac{5}{9} (- {(w^{4})}^{- 2} \frac{d}{d w} [w^{4}]) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.4

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

$\frac{5}{9} (- {(w^{4})}^{- 2} (4 w^{3})) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.5

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

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Step 2.5.1

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

$\frac{5}{9} (- w^{4 \cdot - 2} (4 w^{3})) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.5.2

Multiply $4$ by $- 2$ .

$\frac{5}{9} (- w^{- 8} (4 w^{3})) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.6

Multiply $4$ by $- 1$ .

$\frac{5}{9} (- 4 w^{- 8} w^{3}) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.7

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

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Step 2.7.1

Move $w^{3}$ .

$\frac{5}{9} (- 4 (w^{3} w^{- 8})) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.7.2

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

$\frac{5}{9} (- 4 w^{3 - 8}) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.7.3

Subtract $8$ from $3$ .

$\frac{5}{9} (- 4 w^{- 5}) + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.8

Combine $- 4$ and $\frac{5}{9}$ .

$\frac{- 4 \cdot 5}{9} w^{- 5} + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.9

Multiply $- 4$ by $5$ .

$\frac{- 20}{9} w^{- 5} + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.10

Combine $\frac{- 20}{9}$ and $w^{- 5}$ .

$\frac{- 20 w^{- 5}}{9} + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.11

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

$\frac{- 20}{9 w^{5}} + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 2.12

Move the negative in front of the fraction.

$- \frac{20}{9 w^{5}} + \frac{d}{d w} [5 \sqrt[3]{w}]$

Step 3

Evaluate $\frac{d}{d w} [5 \sqrt[3]{w}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{w}$ as $w^{\frac{1}{3}}$ .

$- \frac{20}{9 w^{5}} + \frac{d}{d w} [5 w^{\frac{1}{3}}]$

Step 3.2

Since $5$ is constant with respect to $w$ , the derivative of $5 w^{\frac{1}{3}}$ with respect to $w$ is $5 \frac{d}{d w} [w^{\frac{1}{3}}]$ .

$- \frac{20}{9 w^{5}} + 5 \frac{d}{d w} [w^{\frac{1}{3}}]$

Step 3.3

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

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{1}{3} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{1}{3} - 1 \cdot \frac{3}{3}})$

Step 3.5

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

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})$

Step 3.6

Combine the numerators over the common denominator.

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{1 - 1 \cdot 3}{3}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $3$ .

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{1 - 3}{3}})$

Step 3.7.2

Subtract $3$ from $1$ .

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{\frac{- 2}{3}})$

Step 3.8

Move the negative in front of the fraction.

$- \frac{20}{9 w^{5}} + 5 (\frac{1}{3} w^{- \frac{2}{3}})$

Step 3.9

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

$- \frac{20}{9 w^{5}} + 5 \frac{w^{- \frac{2}{3}}}{3}$

Step 3.10

Combine $5$ and $\frac{w^{- \frac{2}{3}}}{3}$ .

$- \frac{20}{9 w^{5}} + \frac{5 w^{- \frac{2}{3}}}{3}$

Step 3.11

Move $w^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{20}{9 w^{5}} + \frac{5}{3 w^{\frac{2}{3}}}$