Calculus Examples

$y = 3 x^{\frac{2}{3}} - \frac{2}{x}$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $- 1$ by $3$ .

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

Step 1.2.6.2

Subtract $3$ from $2$ .

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

Step 1.2.7

Move the negative in front of the fraction.

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Multiply $2$ by $3$ .

$\frac{6 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- \frac{2}{x}]$

Step 1.2.11

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

$\frac{6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- \frac{2}{x}]$

Step 1.2.12

Factor $3$ out of $6$ .

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

Step 1.2.13

Cancel the common factors.

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Step 1.2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 1.2.13.2

Cancel the common factor.

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

Step 1.2.13.3

Rewrite the expression.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $- 1$ by $- 2$ .

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.2.3

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 $g (x) = x^{2}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $2$ by $- 2$ .

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

Step 2.2.6

Multiply $2$ by $- 1$ .

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Subtract $4$ from $1$ .

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

Step 2.2.10

Multiply $- 2$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

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 $u_{2}$ as $x^{\frac{1}{3}}$ .

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{3}}$ .

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

Step 2.3.4

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.5.2

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

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

Step 2.3.5.3

Move the negative in front of the fraction.

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

Combine the numerators over the common denominator.

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

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $3$ .

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

Step 2.3.9.2

Subtract $3$ from $1$ .

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

Step 2.3.10

Move the negative in front of the fraction.

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

Step 2.3.11

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

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

Step 2.3.12

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

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

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.

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

Step 2.3.13.2

Combine the numerators over the common denominator.

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

Step 2.3.13.3

Subtract $2$ from $- 2$ .

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

Step 2.3.13.4

Move the negative in front of the fraction.

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

Step 2.3.14

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

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

Step 2.3.15

Multiply $- 1$ by $2$ .

$- 4 x^{- 3} - 2 \frac{1}{3 x^{\frac{4}{3}}}$

Step 2.3.16

Combine $- 2$ and $\frac{1}{3 x^{\frac{4}{3}}}$ .

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

Step 2.3.17

Move the negative in front of the fraction.

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

Step 2.4

Simplify.

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Step 2.4.1

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

$- 4 \frac{1}{x^{3}} - \frac{2}{3 x^{\frac{4}{3}}}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Move the negative in front of the fraction.

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 $g (x) = x^{3}$ .

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$- 4 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{4}{3}}}]$

Step 3.2.6

Multiply $3$ by $- 1$ .

$- 4 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{4}{3}}}]$

Step 3.2.7

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

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Step 3.2.7.1

Move $x^{2}$ .

$- 4 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{4}{3}}}]$

Step 3.2.7.2

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

$- 4 (- 3 x^{2 - 6}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{4}{3}}}]$

Step 3.2.7.3

Subtract $6$ from $2$ .

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

Step 3.2.8

Multiply $- 3$ by $- 4$ .

$12 x^{- 4} + \frac{d}{d x} [- \frac{2}{3 x^{\frac{4}{3}}}]$

Step 3.3

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

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Step 3.3.1

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

$12 x^{- 4} - \frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$

Step 3.3.2

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

$12 x^{- 4} - \frac{2}{3} \frac{d}{d x} [{(x^{\frac{4}{3}})}^{- 1}]$

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 $u_{2}$ as $x^{\frac{4}{3}}$ .

$12 x^{- 4} - \frac{2}{3} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.3.2

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

$12 x^{- 4} - \frac{2}{3} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.3.3

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

$12 x^{- 4} - \frac{2}{3} (- {(x^{\frac{4}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.3.4

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

$12 x^{- 4} - \frac{2}{3} (- {(x^{\frac{4}{3}})}^{- 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$12 x^{- 4} - \frac{2}{3} (- x^{\frac{4}{3} \cdot - 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

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 $- 2$ .

$12 x^{- 4} - \frac{2}{3} (- x^{\frac{4 \cdot - 2}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5.2.2

Multiply $4$ by $- 2$ .

$12 x^{- 4} - \frac{2}{3} (- x^{\frac{- 8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.5.3

Move the negative in front of the fraction.

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.3.6

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

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}))$

Step 3.3.7

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

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 3.3.8

Combine the numerators over the common denominator.

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}))$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $3$ .

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 3}{3}}))$

Step 3.3.9.2

Subtract $3$ from $4$ .

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{1}{3}}))$

Step 3.3.10

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

$12 x^{- 4} - \frac{2}{3} (- x^{- \frac{8}{3}} \frac{4 x^{\frac{1}{3}}}{3})$

Step 3.3.11

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

$12 x^{- 4} - \frac{2}{3} (- \frac{4 x^{\frac{1}{3}} x^{- \frac{8}{3}}}{3})$

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

$12 x^{- 4} - \frac{2}{3} (- \frac{4 (x^{- \frac{8}{3}} x^{\frac{1}{3}})}{3})$

Step 3.3.12.2

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

$12 x^{- 4} - \frac{2}{3} (- \frac{4 x^{- \frac{8}{3} + \frac{1}{3}}}{3})$

Step 3.3.12.3

Combine the numerators over the common denominator.

$12 x^{- 4} - \frac{2}{3} (- \frac{4 x^{\frac{- 8 + 1}{3}}}{3})$

Step 3.3.12.4

Add $- 8$ and $1$ .

$12 x^{- 4} - \frac{2}{3} (- \frac{4 x^{\frac{- 7}{3}}}{3})$

Step 3.3.12.5

Move the negative in front of the fraction.

$12 x^{- 4} - \frac{2}{3} (- \frac{4 x^{- \frac{7}{3}}}{3})$

Step 3.3.13

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

$12 x^{- 4} - \frac{2}{3} (- \frac{4}{3 x^{\frac{7}{3}}})$

Step 3.3.14

Multiply $- 1$ by $- 1$ .

$12 x^{- 4} + 1 (\frac{2}{3}) \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.3.15

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

$12 x^{- 4} + \frac{2}{3} \cdot \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.3.16

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

$12 x^{- 4} + \frac{2 \cdot 4}{3 (3 x^{\frac{7}{3}})}$

Step 3.3.17

Multiply $2$ by $4$ .

$12 x^{- 4} + \frac{8}{3 (3 x^{\frac{7}{3}})}$

Step 3.3.18

Multiply $3$ by $3$ .

$12 x^{- 4} + \frac{8}{9 x^{\frac{7}{3}}}$

Step 3.4

Simplify.

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Step 3.4.1

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

$12 \frac{1}{x^{4}} + \frac{8}{9 x^{\frac{7}{3}}}$

Step 3.4.2

Combine $12$ and $\frac{1}{x^{4}}$ .

$f''' (x) = \frac{12}{x^{4}} + \frac{8}{9 x^{\frac{7}{3}}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $\frac{12}{x^{4}} + \frac{8}{9 x^{\frac{7}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{12}{x^{4}}] + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$ .

$\frac{d}{d x} [\frac{12}{x^{4}}] + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2

Evaluate $\frac{d}{d x} [\frac{12}{x^{4}}]$ .

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Step 4.2.1

Since $12$ is constant with respect to $x$ , the derivative of $\frac{12}{x^{4}}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$12 \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.2

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

$12 \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

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 $g (x) = x^{4}$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{4}$ .

$12 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

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 $n = - 1$ .

$12 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $x^{4}$ .

$12 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.4

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

$12 (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.5

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

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Step 4.2.5.1

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

$12 (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.5.2

Multiply $4$ by $- 2$ .

$12 (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.6

Multiply $4$ by $- 1$ .

$12 (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.7

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

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Step 4.2.7.1

Move $x^{3}$ .

$12 (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.7.2

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

$12 (- 4 x^{3 - 8}) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.7.3

Subtract $8$ from $3$ .

$12 (- 4 x^{- 5}) + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.2.8

Multiply $- 4$ by $12$ .

$- 48 x^{- 5} + \frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{8}{9 x^{\frac{7}{3}}}]$ .

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Step 4.3.1

Since $\frac{8}{9}$ is constant with respect to $x$ , the derivative of $\frac{8}{9 x^{\frac{7}{3}}}$ with respect to $x$ is $\frac{8}{9} \frac{d}{d x} [\frac{1}{x^{\frac{7}{3}}}]$ .

$- 48 x^{- 5} + \frac{8}{9} \frac{d}{d x} [\frac{1}{x^{\frac{7}{3}}}]$

Step 4.3.2

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

$- 48 x^{- 5} + \frac{8}{9} \frac{d}{d x} [{(x^{\frac{7}{3}})}^{- 1}]$

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 $u_{2}$ as $x^{\frac{7}{3}}$ .

$- 48 x^{- 5} + \frac{8}{9} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.3.2

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

$- 48 x^{- 5} + \frac{8}{9} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{7}{3}}$ .

$- 48 x^{- 5} + \frac{8}{9} (- {(x^{\frac{7}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{3}}])$

Step 4.3.4

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

$- 48 x^{- 5} + \frac{8}{9} (- {(x^{\frac{7}{3}})}^{- 2} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$- 48 x^{- 5} + \frac{8}{9} (- x^{\frac{7}{3} \cdot - 2} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

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 $- 2$ .

$- 48 x^{- 5} + \frac{8}{9} (- x^{\frac{7 \cdot - 2}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5.2.2

Multiply $7$ by $- 2$ .

$- 48 x^{- 5} + \frac{8}{9} (- x^{\frac{- 14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.5.3

Move the negative in front of the fraction.

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1}))$

Step 4.3.6

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

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} - 1 \cdot \frac{3}{3}}))$

Step 4.3.7

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

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 4.3.8

Combine the numerators over the common denominator.

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7 - 1 \cdot 3}{3}}))$

Step 4.3.9

Simplify the numerator.

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Step 4.3.9.1

Multiply $- 1$ by $3$ .

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{7 - 3}{3}}))$

Step 4.3.9.2

Subtract $3$ from $7$ .

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} (\frac{7}{3} x^{\frac{4}{3}}))$

Step 4.3.10

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

$- 48 x^{- 5} + \frac{8}{9} (- x^{- \frac{14}{3}} \frac{7 x^{\frac{4}{3}}}{3})$

Step 4.3.11

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

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 x^{\frac{4}{3}} x^{- \frac{14}{3}}}{3})$

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

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 (x^{- \frac{14}{3}} x^{\frac{4}{3}})}{3})$

Step 4.3.12.2

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

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 x^{- \frac{14}{3} + \frac{4}{3}}}{3})$

Step 4.3.12.3

Combine the numerators over the common denominator.

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 x^{\frac{- 14 + 4}{3}}}{3})$

Step 4.3.12.4

Add $- 14$ and $4$ .

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 x^{\frac{- 10}{3}}}{3})$

Step 4.3.12.5

Move the negative in front of the fraction.

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7 x^{- \frac{10}{3}}}{3})$

Step 4.3.13

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

$- 48 x^{- 5} + \frac{8}{9} (- \frac{7}{3 x^{\frac{10}{3}}})$

Step 4.3.14

Multiply $\frac{8}{9}$ by $\frac{7}{3 x^{\frac{10}{3}}}$ .

$- 48 x^{- 5} - \frac{8 \cdot 7}{9 (3 x^{\frac{10}{3}})}$

Step 4.3.15

Multiply $8$ by $7$ .

$- 48 x^{- 5} - \frac{56}{9 (3 x^{\frac{10}{3}})}$

Step 4.3.16

Multiply $3$ by $9$ .

$- 48 x^{- 5} - \frac{56}{27 x^{\frac{10}{3}}}$

Step 4.4

Simplify.

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Step 4.4.1

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

$- 48 \frac{1}{x^{5}} - \frac{56}{27 x^{\frac{10}{3}}}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Combine $- 48$ and $\frac{1}{x^{5}}$ .

$\frac{- 48}{x^{5}} - \frac{56}{27 x^{\frac{10}{3}}}$

Step 4.4.2.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{48}{x^{5}} - \frac{56}{27 x^{\frac{10}{3}}}$