Calculus Examples

$x^{\frac{8}{3}} - 4 x^{\frac{5}{3}}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $x^{\frac{8}{3}} - 4 x^{\frac{5}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{8}{3}}] + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$ .

$\frac{d}{d x} [x^{\frac{8}{3}}] + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2

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

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Step 1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{8}{3}$ .

$\frac{8}{3} x^{\frac{8}{3} - 1} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2.2

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

$\frac{8}{3} x^{\frac{8}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2.3

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

$\frac{8}{3} x^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2.4

Combine the numerators over the common denominator.

$\frac{8}{3} x^{\frac{8 - 1 \cdot 3}{3}} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $3$ .

$\frac{8}{3} x^{\frac{8 - 3}{3}} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.2.5.2

Subtract $3$ from $8$ .

$\frac{8}{3} x^{\frac{5}{3}} + \frac{d}{d x} [- 4 x^{\frac{5}{3}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- 4 x^{\frac{5}{3}}]$ .

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Step 1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{5}{3}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{5}{3}}]$ .

$\frac{8}{3} x^{\frac{5}{3}} - 4 \frac{d}{d x} [x^{\frac{5}{3}}]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{3}$ .

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{5}{3} - 1})$

Step 1.3.3

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

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}})$

Step 1.3.4

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

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.3.5

Combine the numerators over the common denominator.

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}})$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $3$ .

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{5 - 3}{3}})$

Step 1.3.6.2

Subtract $3$ from $5$ .

$\frac{8}{3} x^{\frac{5}{3}} - 4 (\frac{5}{3} x^{\frac{2}{3}})$

Step 1.3.7

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

$\frac{8}{3} x^{\frac{5}{3}} - 4 \frac{5 x^{\frac{2}{3}}}{3}$

Step 1.3.8

Combine $- 4$ and $\frac{5 x^{\frac{2}{3}}}{3}$ .

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

Step 1.3.9

Multiply $5$ by $- 4$ .

$\frac{8}{3} x^{\frac{5}{3}} + \frac{- 20 x^{\frac{2}{3}}}{3}$

Step 1.3.10

Move the negative in front of the fraction.

$\frac{8}{3} x^{\frac{5}{3}} - \frac{20 x^{\frac{2}{3}}}{3}$

Step 1.4

Combine $\frac{8}{3}$ and $x^{\frac{5}{3}}$ .

$f' (x) = \frac{8 x^{\frac{5}{3}}}{3} - \frac{20 x^{\frac{2}{3}}}{3}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{8 x^{\frac{5}{3}}}{3} - \frac{20 x^{\frac{2}{3}}}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{8 x^{\frac{5}{3}}}{3}] + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$ .

$\frac{d}{d x} [\frac{8 x^{\frac{5}{3}}}{3}] + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{8 x^{\frac{5}{3}}}{3}]$ .

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Step 2.2.1

Since $\frac{8}{3}$ is constant with respect to $x$ , the derivative of $\frac{8 x^{\frac{5}{3}}}{3}$ with respect to $x$ is $\frac{8}{3} \frac{d}{d x} [x^{\frac{5}{3}}]$ .

$\frac{8}{3} \frac{d}{d x} [x^{\frac{5}{3}}] + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{3}$ .

$\frac{8}{3} (\frac{5}{3} x^{\frac{5}{3} - 1}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.3

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

$\frac{8}{3} (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.4

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

$\frac{8}{3} (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.5

Combine the numerators over the common denominator.

$\frac{8}{3} (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $3$ .

$\frac{8}{3} (\frac{5}{3} x^{\frac{5 - 3}{3}}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.6.2

Subtract $3$ from $5$ .

$\frac{8}{3} (\frac{5}{3} x^{\frac{2}{3}}) + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.7

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

$\frac{8}{3} \cdot \frac{5 x^{\frac{2}{3}}}{3} + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.8

Multiply $\frac{8}{3}$ by $\frac{5 x^{\frac{2}{3}}}{3}$ .

$\frac{8 (5 x^{\frac{2}{3}})}{3 \cdot 3} + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.9

Multiply $5$ by $8$ .

$\frac{40 x^{\frac{2}{3}}}{3 \cdot 3} + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.2.10

Multiply $3$ by $3$ .

$\frac{40 x^{\frac{2}{3}}}{9} + \frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{20 x^{\frac{2}{3}}}{3}]$ .

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Step 2.3.1

Since $- \frac{20}{3}$ is constant with respect to $x$ , the derivative of $- \frac{20 x^{\frac{2}{3}}}{3}$ with respect to $x$ is $- \frac{20}{3} \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.3.2

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

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 2.3.3

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

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 2.3.4

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

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.3.5

Combine the numerators over the common denominator.

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $3$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 2.3.6.2

Subtract $3$ from $2$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 2.3.7

Move the negative in front of the fraction.

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} (\frac{2}{3} x^{- \frac{1}{3}})$

Step 2.3.8

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

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{20}{3} \cdot \frac{2 x^{- \frac{1}{3}}}{3}$

Step 2.3.9

Multiply $\frac{2 x^{- \frac{1}{3}}}{3}$ by $\frac{20}{3}$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{2 x^{- \frac{1}{3}} \cdot 20}{3 \cdot 3}$

Step 2.3.10

Multiply $20$ by $2$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{40 x^{- \frac{1}{3}}}{3 \cdot 3}$

Step 2.3.11

Multiply $3$ by $3$ .

$\frac{40 x^{\frac{2}{3}}}{9} - \frac{40 x^{- \frac{1}{3}}}{9}$

Step 2.3.12

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

$f'' (x) = \frac{40 x^{\frac{2}{3}}}{9} - \frac{40}{9 x^{\frac{1}{3}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{40 x^{\frac{2}{3}}}{9} - \frac{40}{9 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{40 x^{\frac{2}{3}}}{9}] + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$ .

$\frac{d}{d x} [\frac{40 x^{\frac{2}{3}}}{9}] + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{40 x^{\frac{2}{3}}}{9}]$ .

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Step 3.2.1

Since $\frac{40}{9}$ is constant with respect to $x$ , the derivative of $\frac{40 x^{\frac{2}{3}}}{9}$ with respect to $x$ is $\frac{40}{9} \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$\frac{40}{9} \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.2

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

$\frac{40}{9} (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.3

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

$\frac{40}{9} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.4

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

$\frac{40}{9} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.5

Combine the numerators over the common denominator.

$\frac{40}{9} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 1$ by $3$ .

$\frac{40}{9} (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.6.2

Subtract $3$ from $2$ .

$\frac{40}{9} (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.7

Move the negative in front of the fraction.

$\frac{40}{9} (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.8

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

$\frac{40}{9} \cdot \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.9

Multiply $\frac{40}{9}$ by $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{40 (2 x^{- \frac{1}{3}})}{9 \cdot 3} + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.10

Multiply $2$ by $40$ .

$\frac{80 x^{- \frac{1}{3}}}{9 \cdot 3} + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.11

Multiply $9$ by $3$ .

$\frac{80 x^{- \frac{1}{3}}}{27} + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.2.12

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

$\frac{80}{27 x^{\frac{1}{3}}} + \frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{40}{9 x^{\frac{1}{3}}}]$ .

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Step 3.3.1

Since $- \frac{40}{9}$ is constant with respect to $x$ , the derivative of $- \frac{40}{9 x^{\frac{1}{3}}}$ with respect to $x$ is $- \frac{40}{9} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$

Step 3.3.2

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}]$

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 $g (x) = x^{\frac{1}{3}}$ .

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Step 3.3.3.1

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{3}}])$

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 $n = - 1$ .

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 3.3.3.3

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}])$

Step 3.3.4

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 3.3.5

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

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Step 3.3.5.1

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 3.3.5.2

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 3.3.5.3

Move the negative in front of the fraction.

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 3.3.6

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}))$

Step 3.3.7

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 3.3.8

Combine the numerators over the common denominator.

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}))$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $3$ .

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}}))$

Step 3.3.9.2

Subtract $3$ from $1$ .

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}}))$

Step 3.3.10

Move the negative in front of the fraction.

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}}))$

Step 3.3.11

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3})$

Step 3.3.12

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3})$

Step 3.3.13

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

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Step 3.3.13.1

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3})$

Step 3.3.13.2

Combine the numerators over the common denominator.

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{x^{\frac{- 2 - 2}{3}}}{3})$

Step 3.3.13.3

Subtract $2$ from $- 2$ .

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{x^{\frac{- 4}{3}}}{3})$

Step 3.3.13.4

Move the negative in front of the fraction.

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{x^{- \frac{4}{3}}}{3})$

Step 3.3.14

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

$\frac{80}{27 x^{\frac{1}{3}}} - \frac{40}{9} (- \frac{1}{3 x^{\frac{4}{3}}})$

Step 3.3.15

Multiply $- 1$ by $- 1$ .

$\frac{80}{27 x^{\frac{1}{3}}} + 1 (\frac{40}{9}) \frac{1}{3 x^{\frac{4}{3}}}$

Step 3.3.16

Multiply $\frac{40}{9}$ by $1$ .

$\frac{80}{27 x^{\frac{1}{3}}} + \frac{40}{9} \cdot \frac{1}{3 x^{\frac{4}{3}}}$

Step 3.3.17

Multiply $\frac{40}{9}$ by $\frac{1}{3 x^{\frac{4}{3}}}$ .

$\frac{80}{27 x^{\frac{1}{3}}} + \frac{40}{9 (3 x^{\frac{4}{3}})}$

Step 3.3.18

Multiply $3$ by $9$ .

$f''' (x) = \frac{80}{27 x^{\frac{1}{3}}} + \frac{40}{27 x^{\frac{4}{3}}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $\frac{80}{27 x^{\frac{1}{3}}} + \frac{40}{27 x^{\frac{4}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{80}{27 x^{\frac{1}{3}}}] + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$ .

$\frac{d}{d x} [\frac{80}{27 x^{\frac{1}{3}}}] + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2

Evaluate $\frac{d}{d x} [\frac{80}{27 x^{\frac{1}{3}}}]$ .

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Step 4.2.1

Since $\frac{80}{27}$ is constant with respect to $x$ , the derivative of $\frac{80}{27 x^{\frac{1}{3}}}$ with respect to $x$ is $\frac{80}{27} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}]$ .

$\frac{80}{27} \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}] + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.2

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

$\frac{80}{27} \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}] + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.3

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 4.2.3.1

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

$\frac{80}{27} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.3.2

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

$\frac{80}{27} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.3.3

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

$\frac{80}{27} (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.4

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

$\frac{80}{27} (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.5

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

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Step 4.2.5.1

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

$\frac{80}{27} (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.5.2

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

$\frac{80}{27} (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.5.3

Move the negative in front of the fraction.

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.6

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

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.7

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

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.8

Combine the numerators over the common denominator.

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.9

Simplify the numerator.

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Step 4.2.9.1

Multiply $- 1$ by $3$ .

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.9.2

Subtract $3$ from $1$ .

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.10

Move the negative in front of the fraction.

$\frac{80}{27} (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}})) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.11

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

$\frac{80}{27} (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.12

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

$\frac{80}{27} (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.13

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

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Step 4.2.13.1

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

$\frac{80}{27} (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.13.2

Combine the numerators over the common denominator.

$\frac{80}{27} (- \frac{x^{\frac{- 2 - 2}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.13.3

Subtract $2$ from $- 2$ .

$\frac{80}{27} (- \frac{x^{\frac{- 4}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.13.4

Move the negative in front of the fraction.

$\frac{80}{27} (- \frac{x^{- \frac{4}{3}}}{3}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.14

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

$\frac{80}{27} (- \frac{1}{3 x^{\frac{4}{3}}}) + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.15

Multiply $\frac{80}{27}$ by $\frac{1}{3 x^{\frac{4}{3}}}$ .

$- \frac{80}{27 (3 x^{\frac{4}{3}})} + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.2.16

Multiply $3$ by $27$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{40}{27 x^{\frac{4}{3}}}]$ .

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Step 4.3.1

Since $\frac{40}{27}$ is constant with respect to $x$ , the derivative of $\frac{40}{27 x^{\frac{4}{3}}}$ with respect to $x$ is $\frac{40}{27} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$

Step 4.3.2

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} \frac{d}{d x} [{(x^{\frac{4}{3}})}^{- 1}]$

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 $g (x) = x^{\frac{4}{3}}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{4}{3}}$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 4.3.3.2

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{4}{3}}$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- {(x^{\frac{4}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 4.3.4

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- {(x^{\frac{4}{3}})}^{- 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 4.3.5

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

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Step 4.3.5.1

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{\frac{4}{3} \cdot - 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 4.3.5.2

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

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Step 4.3.5.2.1

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{\frac{4 \cdot - 2}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 4.3.5.2.2

Multiply $4$ by $- 2$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{\frac{- 8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 4.3.5.3

Move the negative in front of the fraction.

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 4.3.6

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}))$

Step 4.3.7

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 4.3.8

Combine the numerators over the common denominator.

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}))$

Step 4.3.9

Simplify the numerator.

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Step 4.3.9.1

Multiply $- 1$ by $3$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 3}{3}}))$

Step 4.3.9.2

Subtract $3$ from $4$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{1}{3}}))$

Step 4.3.10

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- x^{- \frac{8}{3}} \frac{4 x^{\frac{1}{3}}}{3})$

Step 4.3.11

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 x^{\frac{1}{3}} x^{- \frac{8}{3}}}{3})$

Step 4.3.12

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

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Step 4.3.12.1

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 (x^{- \frac{8}{3}} x^{\frac{1}{3}})}{3})$

Step 4.3.12.2

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 x^{- \frac{8}{3} + \frac{1}{3}}}{3})$

Step 4.3.12.3

Combine the numerators over the common denominator.

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 x^{\frac{- 8 + 1}{3}}}{3})$

Step 4.3.12.4

Add $- 8$ and $1$ .

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 x^{\frac{- 7}{3}}}{3})$

Step 4.3.12.5

Move the negative in front of the fraction.

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4 x^{- \frac{7}{3}}}{3})$

Step 4.3.13

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

$- \frac{80}{81 x^{\frac{4}{3}}} + \frac{40}{27} (- \frac{4}{3 x^{\frac{7}{3}}})$

Step 4.3.14

Multiply $\frac{40}{27}$ by $\frac{4}{3 x^{\frac{7}{3}}}$ .

$- \frac{80}{81 x^{\frac{4}{3}}} - \frac{40 \cdot 4}{27 (3 x^{\frac{7}{3}})}$

Step 4.3.15

Multiply $40$ by $4$ .

$- \frac{80}{81 x^{\frac{4}{3}}} - \frac{160}{27 (3 x^{\frac{7}{3}})}$

Step 4.3.16

Multiply $3$ by $27$ .

$f^{4} (x) = - \frac{80}{81 x^{\frac{4}{3}}} - \frac{160}{81 x^{\frac{7}{3}}}$