Calculus Examples

$y = \frac{9}{7 w^{5}} + 9 \sqrt[4]{w}$

Step 1

By the Sum Rule, the derivative of $\frac{9}{7 w^{5}} + 9 \sqrt[4]{w}$ with respect to $w$ is $\frac{d}{d w} [\frac{9}{7 w^{5}}] + \frac{d}{d w} [9 \sqrt[4]{w}]$ .

$\frac{d}{d w} [\frac{9}{7 w^{5}}] + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2

Evaluate $\frac{d}{d w} [\frac{9}{7 w^{5}}]$ .

Tap for more steps...

Step 2.1

Since $\frac{9}{7}$ is constant with respect to $w$ , the derivative of $\frac{9}{7 w^{5}}$ with respect to $w$ is $\frac{9}{7} \frac{d}{d w} [\frac{1}{w^{5}}]$ .

$\frac{9}{7} \frac{d}{d w} [\frac{1}{w^{5}}] + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.2

Rewrite $\frac{1}{w^{5}}$ as ${(w^{5})}^{- 1}$ .

$\frac{9}{7} \frac{d}{d w} [{(w^{5})}^{- 1}] + \frac{d}{d w} [9 \sqrt[4]{w}]$

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 $g (w) = w^{5}$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $w^{5}$ .

$\frac{9}{7} (\frac{d}{d u} [u^{- 1}] \frac{d}{d w} [w^{5}]) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.3.2

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

$\frac{9}{7} (- u^{- 2} \frac{d}{d w} [w^{5}]) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.3.3

Replace all occurrences of $u$ with $w^{5}$ .

$\frac{9}{7} (- {(w^{5})}^{- 2} \frac{d}{d w} [w^{5}]) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 5$ .

$\frac{9}{7} (- {(w^{5})}^{- 2} (5 w^{4})) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.5

Multiply the exponents in ${(w^{5})}^{- 2}$ .

Tap for more steps...

Step 2.5.1

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

$\frac{9}{7} (- w^{5 \cdot - 2} (5 w^{4})) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.5.2

Multiply $5$ by $- 2$ .

$\frac{9}{7} (- w^{- 10} (5 w^{4})) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.6

Multiply $5$ by $- 1$ .

$\frac{9}{7} (- 5 w^{- 10} w^{4}) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.7

Multiply $w^{- 10}$ by $w^{4}$ by adding the exponents.

Tap for more steps...

Step 2.7.1

Move $w^{4}$ .

$\frac{9}{7} (- 5 (w^{4} w^{- 10})) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.7.2

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

$\frac{9}{7} (- 5 w^{4 - 10}) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.7.3

Subtract $10$ from $4$ .

$\frac{9}{7} (- 5 w^{- 6}) + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.8

Combine $- 5$ and $\frac{9}{7}$ .

$\frac{- 5 \cdot 9}{7} w^{- 6} + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.9

Multiply $- 5$ by $9$ .

$\frac{- 45}{7} w^{- 6} + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.10

Combine $\frac{- 45}{7}$ and $w^{- 6}$ .

$\frac{- 45 w^{- 6}}{7} + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.11

Move $w^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 45}{7 w^{6}} + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 2.12

Move the negative in front of the fraction.

$- \frac{45}{7 w^{6}} + \frac{d}{d w} [9 \sqrt[4]{w}]$

Step 3

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

Tap for more steps...

Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{w}$ as $w^{\frac{1}{4}}$ .

$- \frac{45}{7 w^{6}} + \frac{d}{d w} [9 w^{\frac{1}{4}}]$

Step 3.2

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

$- \frac{45}{7 w^{6}} + 9 \frac{d}{d w} [w^{\frac{1}{4}}]$

Step 3.3

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

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{1}{4} - 1})$

Step 3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 3.5

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

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 3.6

Combine the numerators over the common denominator.

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{1 - 1 \cdot 4}{4}})$

Step 3.7

Simplify the numerator.

Tap for more steps...

Step 3.7.1

Multiply $- 1$ by $4$ .

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{1 - 4}{4}})$

Step 3.7.2

Subtract $4$ from $1$ .

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{\frac{- 3}{4}})$

Step 3.8

Move the negative in front of the fraction.

$- \frac{45}{7 w^{6}} + 9 (\frac{1}{4} w^{- \frac{3}{4}})$

Step 3.9

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

$- \frac{45}{7 w^{6}} + 9 \frac{w^{- \frac{3}{4}}}{4}$

Step 3.10

Combine $9$ and $\frac{w^{- \frac{3}{4}}}{4}$ .

$- \frac{45}{7 w^{6}} + \frac{9 w^{- \frac{3}{4}}}{4}$

Step 3.11

Move $w^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{45}{7 w^{6}} + \frac{9}{4 w^{\frac{3}{4}}}$