Calculus Examples

$f (x) = \frac{5}{24} x^{- 2} + \frac{1}{8} x^{2} - \frac{3}{10} x^{5}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{5}{24} x^{- 2} + \frac{1}{8} x^{2} - \frac{3}{10} x^{5}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{24} x^{- 2}] + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$ .

$\frac{d}{d x} [\frac{5}{24} x^{- 2}] + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{5}{24} x^{- 2}]$ .

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Step 1.2.1

Since $\frac{5}{24}$ is constant with respect to $x$ , the derivative of $\frac{5}{24} x^{- 2}$ with respect to $x$ is $\frac{5}{24} \frac{d}{d x} [x^{- 2}]$ .

$\frac{5}{24} \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.2

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

$\frac{5}{24} (- 2 x^{- 3}) + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.3

Combine $- 2$ and $\frac{5}{24}$ .

$\frac{- 2 \cdot 5}{24} x^{- 3} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.4

Multiply $- 2$ by $5$ .

$\frac{- 10}{24} x^{- 3} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.5

Combine $\frac{- 10}{24}$ and $x^{- 3}$ .

$\frac{- 10 x^{- 3}}{24} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.6

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

$\frac{- 10}{24 x^{3}} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.7

Cancel the common factor of $- 10$ and $24$ .

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Step 1.2.7.1

Factor $2$ out of $- 10$ .

$\frac{2 (- 5)}{24 x^{3}} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $2$ out of $24 x^{3}$ .

$\frac{2 (- 5)}{2 (12 x^{3})} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.7.2.2

Cancel the common factor.

$\frac{2 \cdot - 5}{2 (12 x^{3})} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.7.2.3

Rewrite the expression.

$\frac{- 5}{12 x^{3}} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.2.8

Move the negative in front of the fraction.

$- \frac{5}{12 x^{3}} + \frac{d}{d x} [\frac{1}{8} x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{1}{8} x^{2}]$ .

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Step 1.3.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{1}{8} x^{2}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{2}]$ .

$- \frac{5}{12 x^{3}} + \frac{1}{8} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.2

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

$- \frac{5}{12 x^{3}} + \frac{1}{8} (2 x) + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.3

Combine $2$ and $\frac{1}{8}$ .

$- \frac{5}{12 x^{3}} + \frac{2}{8} x + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.4

Combine $\frac{2}{8}$ and $x$ .

$- \frac{5}{12 x^{3}} + \frac{2 x}{8} + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.5

Cancel the common factor of $2$ and $8$ .

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Step 1.3.5.1

Factor $2$ out of $2 x$ .

$- \frac{5}{12 x^{3}} + \frac{2 (x)}{8} + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.5.2

Cancel the common factors.

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Step 1.3.5.2.1

Factor $2$ out of $8$ .

$- \frac{5}{12 x^{3}} + \frac{2 x}{2 \cdot 4} + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.5.2.2

Cancel the common factor.

$- \frac{5}{12 x^{3}} + \frac{2 x}{2 \cdot 4} + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.3.5.2.3

Rewrite the expression.

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{d}{d x} [- \frac{3}{10} x^{5}]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{3}{10} x^{5}]$ .

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Step 1.4.1

Since $- \frac{3}{10}$ is constant with respect to $x$ , the derivative of $- \frac{3}{10} x^{5}$ with respect to $x$ is $- \frac{3}{10} \frac{d}{d x} [x^{5}]$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} - \frac{3}{10} \frac{d}{d x} [x^{5}]$

Step 1.4.2

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

$- \frac{5}{12 x^{3}} + \frac{x}{4} - \frac{3}{10} (5 x^{4})$

Step 1.4.3

Multiply $5$ by $- 1$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} - 5 (\frac{3}{10}) x^{4}$

Step 1.4.4

Combine $- 5$ and $\frac{3}{10}$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{- 5 \cdot 3}{10} x^{4}$

Step 1.4.5

Multiply $- 5$ by $3$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{- 15}{10} x^{4}$

Step 1.4.6

Combine $\frac{- 15}{10}$ and $x^{4}$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{- 15 x^{4}}{10}$

Step 1.4.7

Cancel the common factor of $- 15$ and $10$ .

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Step 1.4.7.1

Factor $5$ out of $- 15 x^{4}$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{5 (- 3 x^{4})}{10}$

Step 1.4.7.2

Cancel the common factors.

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Step 1.4.7.2.1

Factor $5$ out of $10$ .

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{5 (- 3 x^{4})}{5 (2)}$

Step 1.4.7.2.2

Cancel the common factor.

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{5 (- 3 x^{4})}{5 \cdot 2}$

Step 1.4.7.2.3

Rewrite the expression.

$- \frac{5}{12 x^{3}} + \frac{x}{4} + \frac{- 3 x^{4}}{2}$

Step 1.4.8

Move the negative in front of the fraction.

$- \frac{5}{12 x^{3}} + \frac{x}{4} - \frac{3 x^{4}}{2}$

Step 1.5

Reorder terms.

$f' (x) = - \frac{3 x^{4}}{2} + \frac{x}{4} - \frac{5}{12 x^{3}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{3 x^{4}}{2} + \frac{x}{4} - \frac{5}{12 x^{3}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{3 x^{4}}{2}] + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$ .

$\frac{d}{d x} [- \frac{3 x^{4}}{2}] + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2

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

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Step 2.2.1

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

$- \frac{3}{2} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.2

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

$- \frac{3}{2} (4 x^{3}) + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.3

Multiply $4$ by $- 1$ .

$- 4 (\frac{3}{2}) x^{3} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.4

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

$\frac{- 4 \cdot 3}{2} x^{3} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.5

Multiply $- 4$ by $3$ .

$\frac{- 12}{2} x^{3} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.6

Combine $\frac{- 12}{2}$ and $x^{3}$ .

$\frac{- 12 x^{3}}{2} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.7

Cancel the common factor of $- 12$ and $2$ .

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Step 2.2.7.1

Factor $2$ out of $- 12 x^{3}$ .

$\frac{2 (- 6 x^{3})}{2} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 6 x^{3})}{2 (1)} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.7.2.2

Cancel the common factor.

$\frac{2 (- 6 x^{3})}{2 \cdot 1} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.7.2.3

Rewrite the expression.

$\frac{- 6 x^{3}}{1} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.2.7.2.4

Divide $- 6 x^{3}$ by $1$ .

$- 6 x^{3} + \frac{d}{d x} [\frac{x}{4}] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.3

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

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Step 2.3.1

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

$- 6 x^{3} + \frac{1}{4} \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.3.2

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

$- 6 x^{3} + \frac{1}{4} \cdot 1 + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.3.3

Multiply $\frac{1}{4}$ by $1$ .

$- 6 x^{3} + \frac{1}{4} + \frac{d}{d x} [- \frac{5}{12 x^{3}}]$

Step 2.4

Evaluate $\frac{d}{d x} [- \frac{5}{12 x^{3}}]$ .

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Step 2.4.1

Since $- \frac{5}{12}$ is constant with respect to $x$ , the derivative of $- \frac{5}{12 x^{3}}$ with respect to $x$ is $- \frac{5}{12} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 2.4.2

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 2.4.3

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 2.4.3.1

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}])$

Step 2.4.3.2

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- u^{- 2} \frac{d}{d x} [x^{3}])$

Step 2.4.3.3

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 2.4.4

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 2.4.5

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

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Step 2.4.5.1

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- x^{3 \cdot - 2} (3 x^{2}))$

Step 2.4.5.2

Multiply $3$ by $- 2$ .

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- x^{- 6} (3 x^{2}))$

Step 2.4.6

Multiply $3$ by $- 1$ .

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- 3 x^{- 6} x^{2})$

Step 2.4.7

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

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Step 2.4.7.1

Move $x^{2}$ .

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- 3 (x^{2} x^{- 6}))$

Step 2.4.7.2

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

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- 3 x^{2 - 6})$

Step 2.4.7.3

Subtract $6$ from $2$ .

$- 6 x^{3} + \frac{1}{4} - \frac{5}{12} (- 3 x^{- 4})$

Step 2.4.8

Multiply $- 3$ by $- 1$ .

$- 6 x^{3} + \frac{1}{4} + 3 (\frac{5}{12}) x^{- 4}$

Step 2.4.9

Combine $3$ and $\frac{5}{12}$ .

$- 6 x^{3} + \frac{1}{4} + \frac{3 \cdot 5}{12} x^{- 4}$

Step 2.4.10

Multiply $3$ by $5$ .

$- 6 x^{3} + \frac{1}{4} + \frac{15}{12} x^{- 4}$

Step 2.4.11

Combine $\frac{15}{12}$ and $x^{- 4}$ .

$- 6 x^{3} + \frac{1}{4} + \frac{15 x^{- 4}}{12}$

Step 2.4.12

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

$- 6 x^{3} + \frac{1}{4} + \frac{15}{12 x^{4}}$

Step 2.4.13

Cancel the common factor of $15$ and $12$ .

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Step 2.4.13.1

Factor $3$ out of $15$ .

$- 6 x^{3} + \frac{1}{4} + \frac{3 (5)}{12 x^{4}}$

Step 2.4.13.2

Cancel the common factors.

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Step 2.4.13.2.1

Factor $3$ out of $12 x^{4}$ .

$- 6 x^{3} + \frac{1}{4} + \frac{3 (5)}{3 (4 x^{4})}$

Step 2.4.13.2.2

Cancel the common factor.

$- 6 x^{3} + \frac{1}{4} + \frac{3 \cdot 5}{3 (4 x^{4})}$

Step 2.4.13.2.3

Rewrite the expression.

$- 6 x^{3} + \frac{1}{4} + \frac{5}{4 x^{4}}$

Step 2.5

Reorder terms.

$f'' (x) = - 6 x^{3} + \frac{5}{4 x^{4}} + \frac{1}{4}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 6 x^{3} + \frac{5}{4 x^{4}} + \frac{1}{4}$ with respect to $x$ is $\frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [\frac{5}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$ .

$\frac{d}{d x} [- 6 x^{3}] + \frac{d}{d x} [\frac{5}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$

Step 3.2

Evaluate $\frac{d}{d x} [- 6 x^{3}]$ .

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Step 3.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{3}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{3}]$ .

$- 6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{5}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$

Step 3.2.2

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

$- 6 (3 x^{2}) + \frac{d}{d x} [\frac{5}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$

Step 3.2.3

Multiply $3$ by $- 6$ .

$- 18 x^{2} + \frac{d}{d x} [\frac{5}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3

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

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Step 3.3.1

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5}{4 x^{4}}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 18 x^{2} + \frac{5}{4} \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.2

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

$- 18 x^{2} + \frac{5}{4} \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{1}{4}]$

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

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Step 3.3.3.1

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

$- 18 x^{2} + \frac{5}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4}]$

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 $n = - 1$ .

$- 18 x^{2} + \frac{5}{4} (- u^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.3.3

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

$- 18 x^{2} + \frac{5}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.4

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

$- 18 x^{2} + \frac{5}{4} (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.5

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

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Step 3.3.5.1

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

$- 18 x^{2} + \frac{5}{4} (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.5.2

Multiply $4$ by $- 2$ .

$- 18 x^{2} + \frac{5}{4} (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.6

Multiply $4$ by $- 1$ .

$- 18 x^{2} + \frac{5}{4} (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.7

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

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Step 3.3.7.1

Move $x^{3}$ .

$- 18 x^{2} + \frac{5}{4} (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.7.2

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

$- 18 x^{2} + \frac{5}{4} (- 4 x^{3 - 8}) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.7.3

Subtract $8$ from $3$ .

$- 18 x^{2} + \frac{5}{4} (- 4 x^{- 5}) + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.8

Combine $- 4$ and $\frac{5}{4}$ .

$- 18 x^{2} + \frac{- 4 \cdot 5}{4} x^{- 5} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.9

Multiply $- 4$ by $5$ .

$- 18 x^{2} + \frac{- 20}{4} x^{- 5} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.10

Combine $\frac{- 20}{4}$ and $x^{- 5}$ .

$- 18 x^{2} + \frac{- 20 x^{- 5}}{4} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.11

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 18 x^{2} + \frac{- 20}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.12

Cancel the common factor of $- 20$ and $4$ .

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Step 3.3.12.1

Factor $4$ out of $- 20$ .

$- 18 x^{2} + \frac{4 \cdot - 5}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.12.2

Cancel the common factors.

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Step 3.3.12.2.1

Factor $4$ out of $4 x^{5}$ .

$- 18 x^{2} + \frac{4 \cdot - 5}{4 (x^{5})} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.12.2.2

Cancel the common factor.

$- 18 x^{2} + \frac{4 \cdot - 5}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.12.2.3

Rewrite the expression.

$- 18 x^{2} + \frac{- 5}{x^{5}} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.3.13

Move the negative in front of the fraction.

$- 18 x^{2} - \frac{5}{x^{5}} + \frac{d}{d x} [\frac{1}{4}]$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4}$ with respect to $x$ is $0$ .

$- 18 x^{2} - \frac{5}{x^{5}} + 0$

Step 3.4.2

Add $- 18 x^{2} - \frac{5}{x^{5}}$ and $0$ .

$f''' (x) = - 18 x^{2} - \frac{5}{x^{5}}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 18 x^{2} - \frac{5}{x^{5}}$ .

$- 18 x^{2} - \frac{5}{x^{5}}$