Calculus Examples

$y = \frac{{(w^{3} + 4)}^{5}}{8}$

Step 1

Since $\frac{1}{8}$ is constant with respect to $w$ , the derivative of $\frac{{(w^{3} + 4)}^{5}}{8}$ with respect to $w$ is $\frac{1}{8} \frac{d}{d w} [{(w^{3} + 4)}^{5}]$ .

$\frac{1}{8} \frac{d}{d w} [{(w^{3} + 4)}^{5}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d w} [f (g (w))]$ is $f' (g (w)) g' (w)$ where $f (w) = w^{5}$ and $g (w) = w^{3} + 4$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $w^{3} + 4$ .

$\frac{1}{8} (\frac{d}{d u} [u^{5}] \frac{d}{d w} [w^{3} + 4])$

Step 2.2

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

$\frac{1}{8} (5 u^{4} \frac{d}{d w} [w^{3} + 4])$

Step 2.3

Replace all occurrences of $u$ with $w^{3} + 4$ .

$\frac{1}{8} (5 {(w^{3} + 4)}^{4} \frac{d}{d w} [w^{3} + 4])$

Step 3

Differentiate.

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Step 3.1

Combine $5$ and $\frac{1}{8}$ .

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

Step 3.2

Combine fractions.

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Step 3.2.1

Combine ${(w^{3} + 4)}^{4}$ and $\frac{5}{8}$ .

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

Step 3.2.2

Move $5$ to the left of ${(w^{3} + 4)}^{4}$ .

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

Step 3.3

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

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

Step 3.4

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

$\frac{5 {(w^{3} + 4)}^{4}}{8} (3 w^{2} + \frac{d}{d w} [4])$

Step 3.5

Since $4$ is constant with respect to $w$ , the derivative of $4$ with respect to $w$ is $0$ .

$\frac{5 {(w^{3} + 4)}^{4}}{8} (3 w^{2} + 0)$

Step 3.6

Combine fractions.

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Step 3.6.1

Add $3 w^{2}$ and $0$ .

$\frac{5 {(w^{3} + 4)}^{4}}{8} (3 w^{2})$

Step 3.6.2

Combine $3$ and $\frac{5 {(w^{3} + 4)}^{4}}{8}$ .

$\frac{3 (5 {(w^{3} + 4)}^{4})}{8} w^{2}$

Step 3.6.3

Multiply $5$ by $3$ .

$\frac{15 {(w^{3} + 4)}^{4}}{8} w^{2}$

Step 3.6.4

Combine $\frac{15 {(w^{3} + 4)}^{4}}{8}$ and $w^{2}$ .

$\frac{15 {(w^{3} + 4)}^{4} w^{2}}{8}$

Step 4

Simplify the numerator.

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Step 4.1

Use the Binomial Theorem.

$\frac{15 ({(w^{3})}^{4} + 4 {(w^{3})}^{3} \cdot 4 + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply the exponents in ${(w^{3})}^{4}$ .

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Step 4.2.1.1

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

$\frac{15 (w^{3 \cdot 4} + 4 {(w^{3})}^{3} \cdot 4 + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.1.2

Multiply $3$ by $4$ .

$\frac{15 (w^{12} + 4 {(w^{3})}^{3} \cdot 4 + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.2

Multiply the exponents in ${(w^{3})}^{3}$ .

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Step 4.2.2.1

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

$\frac{15 (w^{12} + 4 w^{3 \cdot 3} \cdot 4 + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.2.2

Multiply $3$ by $3$ .

$\frac{15 (w^{12} + 4 w^{9} \cdot 4 + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.3

Multiply $4$ by $4$ .

$\frac{15 (w^{12} + 16 w^{9} + 6 {(w^{3})}^{2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.4

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

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Step 4.2.4.1

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

$\frac{15 (w^{12} + 16 w^{9} + 6 w^{3 \cdot 2} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.4.2

Multiply $3$ by $2$ .

$\frac{15 (w^{12} + 16 w^{9} + 6 w^{6} \cdot 4^{2} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.5

Raise $4$ to the power of $2$ .

$\frac{15 (w^{12} + 16 w^{9} + 6 w^{6} \cdot 16 + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.6

Multiply $16$ by $6$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 4 w^{3} \cdot 4^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.7

Multiply $4$ by $4^{3}$ by adding the exponents.

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Step 4.2.7.1

Move $4^{3}$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 4^{3} \cdot 4 w^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.7.2

Multiply $4^{3}$ by $4$ .

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Step 4.2.7.2.1

Raise $4$ to the power of $1$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 4^{3} \cdot 4^{1} w^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.7.2.2

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

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 4^{3 + 1} w^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.7.3

Add $3$ and $1$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 4^{4} w^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.8

Raise $4$ to the power of $4$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 256 w^{3} + 4^{4}) w^{2}}{8}$

Step 4.2.9

Raise $4$ to the power of $4$ .

$\frac{15 (w^{12} + 16 w^{9} + 96 w^{6} + 256 w^{3} + 256) w^{2}}{8}$

Step 4.3

Apply the distributive property.

$\frac{(15 w^{12} + 15 (16 w^{9}) + 15 (96 w^{6}) + 15 (256 w^{3}) + 15 \cdot 256) w^{2}}{8}$

Step 4.4

Simplify.

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Step 4.4.1

Multiply $16$ by $15$ .

$\frac{(15 w^{12} + 240 w^{9} + 15 (96 w^{6}) + 15 (256 w^{3}) + 15 \cdot 256) w^{2}}{8}$

Step 4.4.2

Multiply $96$ by $15$ .

$\frac{(15 w^{12} + 240 w^{9} + 1440 w^{6} + 15 (256 w^{3}) + 15 \cdot 256) w^{2}}{8}$

Step 4.4.3

Multiply $256$ by $15$ .

$\frac{(15 w^{12} + 240 w^{9} + 1440 w^{6} + 3840 w^{3} + 15 \cdot 256) w^{2}}{8}$

Step 4.4.4

Multiply $15$ by $256$ .

$\frac{(15 w^{12} + 240 w^{9} + 1440 w^{6} + 3840 w^{3} + 3840) w^{2}}{8}$

Step 4.5

Apply the distributive property.

$\frac{15 w^{12} w^{2} + 240 w^{9} w^{2} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6

Simplify.

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Step 4.6.1

Multiply $w^{12}$ by $w^{2}$ by adding the exponents.

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Step 4.6.1.1

Move $w^{2}$ .

$\frac{15 (w^{2} w^{12}) + 240 w^{9} w^{2} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.1.2

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

$\frac{15 w^{2 + 12} + 240 w^{9} w^{2} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.1.3

Add $2$ and $12$ .

$\frac{15 w^{14} + 240 w^{9} w^{2} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.2

Multiply $w^{9}$ by $w^{2}$ by adding the exponents.

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Step 4.6.2.1

Move $w^{2}$ .

$\frac{15 w^{14} + 240 (w^{2} w^{9}) + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.2.2

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

$\frac{15 w^{14} + 240 w^{2 + 9} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.2.3

Add $2$ and $9$ .

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{6} w^{2} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.3

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

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Step 4.6.3.1

Move $w^{2}$ .

$\frac{15 w^{14} + 240 w^{11} + 1440 (w^{2} w^{6}) + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.3.2

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

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{2 + 6} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.3.3

Add $2$ and $6$ .

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{8} + 3840 w^{3} w^{2} + 3840 w^{2}}{8}$

Step 4.6.4

Multiply $w^{3}$ by $w^{2}$ by adding the exponents.

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Step 4.6.4.1

Move $w^{2}$ .

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{8} + 3840 (w^{2} w^{3}) + 3840 w^{2}}{8}$

Step 4.6.4.2

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

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{8} + 3840 w^{2 + 3} + 3840 w^{2}}{8}$

Step 4.6.4.3

Add $2$ and $3$ .

$\frac{15 w^{14} + 240 w^{11} + 1440 w^{8} + 3840 w^{5} + 3840 w^{2}}{8}$