Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

Multiply $3$ by $- 2$ .

$5 x^{4} - 2 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.2.6

Multiply $3$ by $- 1$ .

$5 x^{4} - 2 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.2.7

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

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Step 1.2.7.1

Move $x^{2}$ .

$5 x^{4} - 2 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.2.7.2

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

$5 x^{4} - 2 (- 3 x^{2 - 6}) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.2.7.3

Subtract $6$ from $2$ .

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

Step 1.2.8

Multiply $- 3$ by $- 2$ .

$5 x^{4} + 6 x^{- 4} + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.3

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

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Step 1.3.1

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

$5 x^{4} + 6 x^{- 4} - 4 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.3.2

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

$5 x^{4} + 6 x^{- 4} - 4 \cdot 1 + \frac{d}{d x} [2]$

Step 1.3.3

Multiply $- 4$ by $1$ .

$5 x^{4} + 6 x^{- 4} - 4 + \frac{d}{d x} [2]$

Step 1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$5 x^{4} + 6 x^{- 4} - 4 + 0$

Step 1.5

Simplify.

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Step 1.5.1

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

$5 x^{4} + 6 \frac{1}{x^{4}} - 4 + 0$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine $6$ and $\frac{1}{x^{4}}$ .

$5 x^{4} + \frac{6}{x^{4}} - 4 + 0$

Step 1.5.2.2

Add $5 x^{4} + \frac{6}{x^{4}} - 4$ and $0$ .

$f' (x) = 5 x^{4} + \frac{6}{x^{4}} - 4$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{4} + \frac{6}{x^{4}} - 4$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [\frac{6}{x^{4}}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [\frac{6}{x^{4}}] + \frac{d}{d x} [- 4]$

Step 2.2

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

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Step 2.2.1

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

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{6}{x^{4}}] + \frac{d}{d x} [- 4]$

Step 2.2.2

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

$5 (4 x^{3}) + \frac{d}{d x} [\frac{6}{x^{4}}] + \frac{d}{d x} [- 4]$

Step 2.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [\frac{6}{x^{4}}] + \frac{d}{d x} [- 4]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{6}{x^{4}}]$ .

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Step 2.3.1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6}{x^{4}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$20 x^{3} + 6 \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [- 4]$

Step 2.3.2

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

$20 x^{3} + 6 \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [- 4]$

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

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Step 2.3.3.1

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

$20 x^{3} + 6 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [- 4]$

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 $n = - 1$ .

$20 x^{3} + 6 (- u^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [- 4]$

Step 2.3.3.3

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

$20 x^{3} + 6 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [- 4]$

Step 2.3.4

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

$20 x^{3} + 6 (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [- 4]$

Step 2.3.5

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

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Step 2.3.5.1

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

$20 x^{3} + 6 (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [- 4]$

Step 2.3.5.2

Multiply $4$ by $- 2$ .

$20 x^{3} + 6 (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [- 4]$

Step 2.3.6

Multiply $4$ by $- 1$ .

$20 x^{3} + 6 (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [- 4]$

Step 2.3.7

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

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Step 2.3.7.1

Move $x^{3}$ .

$20 x^{3} + 6 (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [- 4]$

Step 2.3.7.2

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

$20 x^{3} + 6 (- 4 x^{3 - 8}) + \frac{d}{d x} [- 4]$

Step 2.3.7.3

Subtract $8$ from $3$ .

$20 x^{3} + 6 (- 4 x^{- 5}) + \frac{d}{d x} [- 4]$

Step 2.3.8

Multiply $- 4$ by $6$ .

$20 x^{3} - 24 x^{- 5} + \frac{d}{d x} [- 4]$

Step 2.4

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

$20 x^{3} - 24 x^{- 5} + 0$

Step 2.5

Simplify.

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Step 2.5.1

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

$20 x^{3} - 24 \frac{1}{x^{5}} + 0$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $- 24$ and $\frac{1}{x^{5}}$ .

$20 x^{3} + \frac{- 24}{x^{5}} + 0$

Step 2.5.2.2

Move the negative in front of the fraction.

$20 x^{3} - \frac{24}{x^{5}} + 0$

Step 2.5.2.3

Add $20 x^{3} - \frac{24}{x^{5}}$ and $0$ .

$f'' (x) = 20 x^{3} - \frac{24}{x^{5}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $20 x^{3} - \frac{24}{x^{5}}$ with respect to $x$ is $\frac{d}{d x} [20 x^{3}] + \frac{d}{d x} [- \frac{24}{x^{5}}]$ .

$\frac{d}{d x} [20 x^{3}] + \frac{d}{d x} [- \frac{24}{x^{5}}]$

Step 3.2

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

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Step 3.2.1

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

$20 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{24}{x^{5}}]$

Step 3.2.2

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

$20 (3 x^{2}) + \frac{d}{d x} [- \frac{24}{x^{5}}]$

Step 3.2.3

Multiply $3$ by $20$ .

$60 x^{2} + \frac{d}{d x} [- \frac{24}{x^{5}}]$

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{24}{x^{5}}]$ .

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Step 3.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- \frac{24}{x^{5}}$ with respect to $x$ is $- 24 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$60 x^{2} - 24 \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 3.3.2

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

$60 x^{2} - 24 \frac{d}{d x} [{(x^{5})}^{- 1}]$

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

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Step 3.3.3.1

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

$60 x^{2} - 24 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}])$

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 $n = - 1$ .

$60 x^{2} - 24 (- u^{- 2} \frac{d}{d x} [x^{5}])$

Step 3.3.3.3

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

$60 x^{2} - 24 (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}])$

Step 3.3.4

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

$60 x^{2} - 24 (- {(x^{5})}^{- 2} (5 x^{4}))$

Step 3.3.5

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

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Step 3.3.5.1

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

$60 x^{2} - 24 (- x^{5 \cdot - 2} (5 x^{4}))$

Step 3.3.5.2

Multiply $5$ by $- 2$ .

$60 x^{2} - 24 (- x^{- 10} (5 x^{4}))$

Step 3.3.6

Multiply $5$ by $- 1$ .

$60 x^{2} - 24 (- 5 x^{- 10} x^{4})$

Step 3.3.7

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

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Step 3.3.7.1

Move $x^{4}$ .

$60 x^{2} - 24 (- 5 (x^{4} x^{- 10}))$

Step 3.3.7.2

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

$60 x^{2} - 24 (- 5 x^{4 - 10})$

Step 3.3.7.3

Subtract $10$ from $4$ .

$60 x^{2} - 24 (- 5 x^{- 6})$

Step 3.3.8

Multiply $- 5$ by $- 24$ .

$60 x^{2} + 120 x^{- 6}$

Step 3.4

Simplify.

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Step 3.4.1

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

$60 x^{2} + 120 \frac{1}{x^{6}}$

Step 3.4.2

Combine $120$ and $\frac{1}{x^{6}}$ .

$f''' (x) = 60 x^{2} + \frac{120}{x^{6}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $60 x^{2} + \frac{120}{x^{6}}$ with respect to $x$ is $\frac{d}{d x} [60 x^{2}] + \frac{d}{d x} [\frac{120}{x^{6}}]$ .

$\frac{d}{d x} [60 x^{2}] + \frac{d}{d x} [\frac{120}{x^{6}}]$

Step 4.2

Evaluate $\frac{d}{d x} [60 x^{2}]$ .

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Step 4.2.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{2}$ with respect to $x$ is $60 \frac{d}{d x} [x^{2}]$ .

$60 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{120}{x^{6}}]$

Step 4.2.2

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

$60 (2 x) + \frac{d}{d x} [\frac{120}{x^{6}}]$

Step 4.2.3

Multiply $2$ by $60$ .

$120 x + \frac{d}{d x} [\frac{120}{x^{6}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{120}{x^{6}}]$ .

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Step 4.3.1

Since $120$ is constant with respect to $x$ , the derivative of $\frac{120}{x^{6}}$ with respect to $x$ is $120 \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$120 x + 120 \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 4.3.2

Rewrite $\frac{1}{x^{6}}$ as ${(x^{6})}^{- 1}$ .

$120 x + 120 \frac{d}{d x} [{(x^{6})}^{- 1}]$

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 $g (x) = x^{6}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$120 x + 120 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{6}])$

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 $n = - 1$ .

$120 x + 120 (- u^{- 2} \frac{d}{d x} [x^{6}])$

Step 4.3.3.3

Replace all occurrences of $u$ with $x^{6}$ .

$120 x + 120 (- {(x^{6})}^{- 2} \frac{d}{d x} [x^{6}])$

Step 4.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$120 x + 120 (- {(x^{6})}^{- 2} (6 x^{5}))$

Step 4.3.5

Multiply the exponents in ${(x^{6})}^{- 2}$ .

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Step 4.3.5.1

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

$120 x + 120 (- x^{6 \cdot - 2} (6 x^{5}))$

Step 4.3.5.2

Multiply $6$ by $- 2$ .

$120 x + 120 (- x^{- 12} (6 x^{5}))$

Step 4.3.6

Multiply $6$ by $- 1$ .

$120 x + 120 (- 6 x^{- 12} x^{5})$

Step 4.3.7

Multiply $x^{- 12}$ by $x^{5}$ by adding the exponents.

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Step 4.3.7.1

Move $x^{5}$ .

$120 x + 120 (- 6 (x^{5} x^{- 12}))$

Step 4.3.7.2

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

$120 x + 120 (- 6 x^{5 - 12})$

Step 4.3.7.3

Subtract $12$ from $5$ .

$120 x + 120 (- 6 x^{- 7})$

Step 4.3.8

Multiply $- 6$ by $120$ .

$120 x - 720 x^{- 7}$

Step 4.4

Simplify.

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Step 4.4.1

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

$120 x - 720 \frac{1}{x^{7}}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Combine $- 720$ and $\frac{1}{x^{7}}$ .

$120 x + \frac{- 720}{x^{7}}$

Step 4.4.2.2

Move the negative in front of the fraction.

$f^{4} (x) = 120 x - \frac{720}{x^{7}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $120 x - \frac{720}{x^{7}}$ .

$120 x - \frac{720}{x^{7}}$