Calculus Examples

$f (x) = \sqrt[3]{x^{2}} + 3 x^{4} - 5$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\sqrt[3]{x^{2}} + 3 x^{4} - 5$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x^{2}}] + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [\sqrt[3]{x^{2}}] + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2

Evaluate $\frac{d}{d x} [\sqrt[3]{x^{2}}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} - 1} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.3

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

$\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.4

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

$\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} x^{\frac{2 - 3}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.6.2

Subtract $3$ from $2$ .

$\frac{2}{3} x^{\frac{- 1}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.2.7

Move the negative in front of the fraction.

$\frac{2}{3} x^{- \frac{1}{3}} + \frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [- 5]$

Step 1.3

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

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Step 1.3.1

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

$\frac{2}{3} x^{- \frac{1}{3}} + 3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5]$

Step 1.3.2

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

$\frac{2}{3} x^{- \frac{1}{3}} + 3 (4 x^{3}) + \frac{d}{d x} [- 5]$

Step 1.3.3

Multiply $4$ by $3$ .

$\frac{2}{3} x^{- \frac{1}{3}} + 12 x^{3} + \frac{d}{d x} [- 5]$

Step 1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{2}{3} x^{- \frac{1}{3}} + 12 x^{3} + 0$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} + 12 x^{3} + 0$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Multiply $\frac{2}{3}$ by $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{2}{3 x^{\frac{1}{3}}} + 12 x^{3} + 0$

Step 1.5.2.2

Add $\frac{2}{3 x^{\frac{1}{3}}} + 12 x^{3}$ and $0$ .

$\frac{2}{3 x^{\frac{1}{3}}} + 12 x^{3}$

Step 1.5.3

Reorder terms.

$f' (x) = 12 x^{3} + \frac{2}{3 x^{\frac{1}{3}}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $12 x^{3} + \frac{2}{3 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$ .

$\frac{d}{d x} [12 x^{3}] + \frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$

Step 2.2

Evaluate $\frac{d}{d x} [12 x^{3}]$ .

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Step 2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{3}$ with respect to $x$ is $12 \frac{d}{d x} [x^{3}]$ .

$12 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$

Step 2.2.2

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

$12 (3 x^{2}) + \frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$

Step 2.2.3

Multiply $3$ by $12$ .

$36 x^{2} + \frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{2}{3 x^{\frac{1}{3}}}]$ .

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Step 2.3.1

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

$36 x^{2} + \frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$

Step 2.3.2

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

$36 x^{2} + \frac{2}{3} \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}]$

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{3}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

$36 x^{2} + \frac{2}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 2.3.3.2

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

$36 x^{2} + \frac{2}{3} (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

$36 x^{2} + \frac{2}{3} (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 2.3.4

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

$36 x^{2} + \frac{2}{3} (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 2}$ .

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Step 2.3.5.1

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

$36 x^{2} + \frac{2}{3} (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.3.5.2

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

$36 x^{2} + \frac{2}{3} (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.3.5.3

Move the negative in front of the fraction.

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 2.3.6

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

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}))$

Step 2.3.7

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

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $3$ .

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}}))$

Step 2.3.9.2

Subtract $3$ from $1$ .

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}}))$

Step 2.3.10

Move the negative in front of the fraction.

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}}))$

Step 2.3.11

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

$36 x^{2} + \frac{2}{3} (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3})$

Step 2.3.12

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

$36 x^{2} + \frac{2}{3} (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3})$

Step 2.3.13

Multiply $x^{- \frac{2}{3}}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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Step 2.3.13.1

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

$36 x^{2} + \frac{2}{3} (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3})$

Step 2.3.13.2

Combine the numerators over the common denominator.

$36 x^{2} + \frac{2}{3} (- \frac{x^{\frac{- 2 - 2}{3}}}{3})$

Step 2.3.13.3

Subtract $2$ from $- 2$ .

$36 x^{2} + \frac{2}{3} (- \frac{x^{\frac{- 4}{3}}}{3})$

Step 2.3.13.4

Move the negative in front of the fraction.

$36 x^{2} + \frac{2}{3} (- \frac{x^{- \frac{4}{3}}}{3})$

Step 2.3.14

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

$36 x^{2} + \frac{2}{3} (- \frac{1}{3 x^{\frac{4}{3}}})$

Step 2.3.15

Multiply $\frac{2}{3}$ by $\frac{1}{3 x^{\frac{4}{3}}}$ .

$36 x^{2} - \frac{2}{3 (3 x^{\frac{4}{3}})}$

Step 2.3.16

Multiply $3$ by $3$ .

$f'' (x) = 36 x^{2} - \frac{2}{9 x^{\frac{4}{3}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $36 x^{2} - \frac{2}{9 x^{\frac{4}{3}}}$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$

Step 3.2

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 3.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$

Step 3.2.2

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

$36 (2 x) + \frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$

Step 3.2.3

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{2}{9 x^{\frac{4}{3}}}]$ .

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Step 3.3.1

Since $- \frac{2}{9}$ is constant with respect to $x$ , the derivative of $- \frac{2}{9 x^{\frac{4}{3}}}$ with respect to $x$ is $- \frac{2}{9} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$ .

$72 x - \frac{2}{9} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{\frac{4}{3}}}$ as ${(x^{\frac{4}{3}})}^{- 1}$ .

$72 x - \frac{2}{9} \frac{d}{d x} [{(x^{\frac{4}{3}})}^{- 1}]$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{4}{3}}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{4}{3}}$ .

$72 x - \frac{2}{9} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.3.2

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

$72 x - \frac{2}{9} (- u^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.3.3

Replace all occurrences of $u$ with $x^{\frac{4}{3}}$ .

$72 x - \frac{2}{9} (- {(x^{\frac{4}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$72 x - \frac{2}{9} (- {(x^{\frac{4}{3}})}^{- 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5

Multiply the exponents in ${(x^{\frac{4}{3}})}^{- 2}$ .

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Step 3.3.5.1

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

$72 x - \frac{2}{9} (- x^{\frac{4}{3} \cdot - 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5.2

Multiply $\frac{4}{3} \cdot - 2$ .

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Step 3.3.5.2.1

Combine $\frac{4}{3}$ and $- 2$ .

$72 x - \frac{2}{9} (- x^{\frac{4 \cdot - 2}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5.2.2

Multiply $4$ by $- 2$ .

$72 x - \frac{2}{9} (- x^{\frac{- 8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5.3

Move the negative in front of the fraction.

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.6

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

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}))$

Step 3.3.7

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

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 3.3.8

Combine the numerators over the common denominator.

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}))$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $3$ .

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 3}{3}}))$

Step 3.3.9.2

Subtract $3$ from $4$ .

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{1}{3}}))$

Step 3.3.10

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

$72 x - \frac{2}{9} (- x^{- \frac{8}{3}} \frac{4 x^{\frac{1}{3}}}{3})$

Step 3.3.11

Combine $\frac{4 x^{\frac{1}{3}}}{3}$ and $x^{- \frac{8}{3}}$ .

$72 x - \frac{2}{9} (- \frac{4 x^{\frac{1}{3}} x^{- \frac{8}{3}}}{3})$

Step 3.3.12

Multiply $x^{\frac{1}{3}}$ by $x^{- \frac{8}{3}}$ by adding the exponents.

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Step 3.3.12.1

Move $x^{- \frac{8}{3}}$ .

$72 x - \frac{2}{9} (- \frac{4 (x^{- \frac{8}{3}} x^{\frac{1}{3}})}{3})$

Step 3.3.12.2

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

$72 x - \frac{2}{9} (- \frac{4 x^{- \frac{8}{3} + \frac{1}{3}}}{3})$

Step 3.3.12.3

Combine the numerators over the common denominator.

$72 x - \frac{2}{9} (- \frac{4 x^{\frac{- 8 + 1}{3}}}{3})$

Step 3.3.12.4

Add $- 8$ and $1$ .

$72 x - \frac{2}{9} (- \frac{4 x^{\frac{- 7}{3}}}{3})$

Step 3.3.12.5

Move the negative in front of the fraction.

$72 x - \frac{2}{9} (- \frac{4 x^{- \frac{7}{3}}}{3})$

Step 3.3.13

Move $x^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$72 x - \frac{2}{9} (- \frac{4}{3 x^{\frac{7}{3}}})$

Step 3.3.14

Multiply $- 1$ by $- 1$ .

$72 x + 1 (\frac{2}{9}) \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.3.15

Multiply $\frac{2}{9}$ by $1$ .

$72 x + \frac{2}{9} \cdot \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.3.16

Multiply $\frac{2}{9}$ by $\frac{4}{3 x^{\frac{7}{3}}}$ .

$72 x + \frac{2 \cdot 4}{9 (3 x^{\frac{7}{3}})}$

Step 3.3.17

Multiply $2$ by $4$ .

$72 x + \frac{8}{9 (3 x^{\frac{7}{3}})}$

Step 3.3.18

Multiply $3$ by $9$ .

$f''' (x) = 72 x + \frac{8}{27 x^{\frac{7}{3}}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $72 x + \frac{8}{27 x^{\frac{7}{3}}}$ with respect to $x$ is $\frac{d}{d x} [72 x] + \frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$ .

$\frac{d}{d x} [72 x] + \frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$

Step 4.2

Evaluate $\frac{d}{d x} [72 x]$ .

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Step 4.2.1

Since $72$ is constant with respect to $x$ , the derivative of $72 x$ with respect to $x$ is $72 \frac{d}{d x} [x]$ .

$72 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$

Step 4.2.2

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

$72 \cdot 1 + \frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$

Step 4.2.3

Multiply $72$ by $1$ .

$72 + \frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{8}{27 x^{\frac{7}{3}}}]$ .

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Step 4.3.1

Since $\frac{8}{27}$ is constant with respect to $x$ , the derivative of $\frac{8}{27 x^{\frac{7}{3}}}$ with respect to $x$ is $\frac{8}{27} \frac{d}{d x} [\frac{1}{x^{\frac{7}{3}}}]$ .

$72 + \frac{8}{27} \frac{d}{d x} [\frac{1}{x^{\frac{7}{3}}}]$

Step 4.3.2

Rewrite $\frac{1}{x^{\frac{7}{3}}}$ as ${(x^{\frac{7}{3}})}^{- 1}$ .

$72 + \frac{8}{27} \frac{d}{d x} [{(x^{\frac{7}{3}})}^{- 1}]$

Step 4.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{7}{3}}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{7}{3}}$ .

$72 + \frac{8}{27} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.3.2

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

$72 + \frac{8}{27} (- u^{- 2} \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.3.3

Replace all occurrences of $u$ with $x^{\frac{7}{3}}$ .

$72 + \frac{8}{27} (- {(x^{\frac{7}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{3}$ .

$72 + \frac{8}{27} (- {(x^{\frac{7}{3}})}^{- 2} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5

Multiply the exponents in ${(x^{\frac{7}{3}})}^{- 2}$ .

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Step 4.3.5.1

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

$72 + \frac{8}{27} (- x^{\frac{7}{3} \cdot - 2} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5.2

Multiply $\frac{7}{3} \cdot - 2$ .

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Step 4.3.5.2.1

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

$72 + \frac{8}{27} (- x^{\frac{7 \cdot - 2}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5.2.2

Multiply $7$ by $- 2$ .

$72 + \frac{8}{27} (- x^{\frac{- 14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5.3

Move the negative in front of the fraction.

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.6

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

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1 \cdot \frac{3}{3}}))$

Step 4.3.7

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

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 4.3.8

Combine the numerators over the common denominator.

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7 - 1 \cdot 3}{3}}))$

Step 4.3.9

Simplify the numerator.

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Step 4.3.9.1

Multiply $- 1$ by $3$ .

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7 - 3}{3}}))$

Step 4.3.9.2

Subtract $3$ from $7$ .

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{4}{3}}))$

Step 4.3.10

Combine $\frac{7}{3}$ and $x^{\frac{4}{3}}$ .

$72 + \frac{8}{27} (- x^{- \frac{14}{3}} \frac{7 x^{\frac{4}{3}}}{3})$

Step 4.3.11

Combine $\frac{7 x^{\frac{4}{3}}}{3}$ and $x^{- \frac{14}{3}}$ .

$72 + \frac{8}{27} (- \frac{7 x^{\frac{4}{3}} x^{- \frac{14}{3}}}{3})$

Step 4.3.12

Multiply $x^{\frac{4}{3}}$ by $x^{- \frac{14}{3}}$ by adding the exponents.

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Step 4.3.12.1

Move $x^{- \frac{14}{3}}$ .

$72 + \frac{8}{27} (- \frac{7 (x^{- \frac{14}{3}} x^{\frac{4}{3}})}{3})$

Step 4.3.12.2

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

$72 + \frac{8}{27} (- \frac{7 x^{- \frac{14}{3} + \frac{4}{3}}}{3})$

Step 4.3.12.3

Combine the numerators over the common denominator.

$72 + \frac{8}{27} (- \frac{7 x^{\frac{- 14 + 4}{3}}}{3})$

Step 4.3.12.4

Add $- 14$ and $4$ .

$72 + \frac{8}{27} (- \frac{7 x^{\frac{- 10}{3}}}{3})$

Step 4.3.12.5

Move the negative in front of the fraction.

$72 + \frac{8}{27} (- \frac{7 x^{- \frac{10}{3}}}{3})$

Step 4.3.13

Move $x^{- \frac{10}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$72 + \frac{8}{27} (- \frac{7}{3 x^{\frac{10}{3}}})$

Step 4.3.14

Multiply $\frac{8}{27}$ by $\frac{7}{3 x^{\frac{10}{3}}}$ .

$72 - \frac{8 \cdot 7}{27 (3 x^{\frac{10}{3}})}$

Step 4.3.15

Multiply $8$ by $7$ .

$72 - \frac{56}{27 (3 x^{\frac{10}{3}})}$

Step 4.3.16

Multiply $3$ by $27$ .

$f^{4} (x) = 72 - \frac{56}{81 x^{\frac{10}{3}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $72 - \frac{56}{81 x^{\frac{10}{3}}}$ .

$72 - \frac{56}{81 x^{\frac{10}{3}}}$