Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 3$ by $1$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{5}{8} x^{- 2}]$ .

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Step 1.3.1

Since $- \frac{5}{8}$ is constant with respect to $x$ , the derivative of $- \frac{5}{8} x^{- 2}$ with respect to $x$ is $- \frac{5}{8} \frac{d}{d x} [x^{- 2}]$ .

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 2$ by $- 1$ .

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

Step 1.3.4

Combine $2$ and $\frac{5}{8}$ .

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

Step 1.3.5

Multiply $2$ by $5$ .

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

Step 1.3.6

Combine $\frac{10}{8}$ and $x^{- 3}$ .

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

Step 1.3.7

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

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

Step 1.3.8

Cancel the common factor of $10$ and $8$ .

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Step 1.3.8.1

Factor $2$ out of $10$ .

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

Step 1.3.8.2

Cancel the common factors.

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Step 1.3.8.2.1

Factor $2$ out of $8 x^{3}$ .

$- 3 + \frac{2 (5)}{2 (4 x^{3})} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.3.8.2.2

Cancel the common factor.

$- 3 + \frac{2 \cdot 5}{2 (4 x^{3})} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.3.8.2.3

Rewrite the expression.

$- 3 + \frac{5}{4 x^{3}} + \frac{d}{d x} [\frac{1}{2} x^{3}] + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.4

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

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Step 1.4.1

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

$- 3 + \frac{5}{4 x^{3}} + \frac{1}{2} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.4.2

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

$- 3 + \frac{5}{4 x^{3}} + \frac{1}{2} (3 x^{2}) + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.4.3

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

$- 3 + \frac{5}{4 x^{3}} + \frac{3}{2} x^{2} + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.4.4

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

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} + \frac{d}{d x} [- \frac{1}{10} x^{5}]$

Step 1.5

Evaluate $\frac{d}{d x} [- \frac{1}{10} x^{5}]$ .

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Step 1.5.1

Since $- \frac{1}{10}$ is constant with respect to $x$ , the derivative of $- \frac{1}{10} x^{5}$ with respect to $x$ is $- \frac{1}{10} \frac{d}{d x} [x^{5}]$ .

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} - \frac{1}{10} \frac{d}{d x} [x^{5}]$

Step 1.5.2

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

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} - \frac{1}{10} (5 x^{4})$

Step 1.5.3

Multiply $5$ by $- 1$ .

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} - 5 (\frac{1}{10}) x^{4}$

Step 1.5.4

Combine $- 5$ and $\frac{1}{10}$ .

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} + \frac{- 5}{10} x^{4}$

Step 1.5.5

Combine $\frac{- 5}{10}$ and $x^{4}$ .

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} + \frac{- 5 x^{4}}{10}$

Step 1.5.6

Cancel the common factor of $- 5$ and $10$ .

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Step 1.5.6.1

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

$- 3 + \frac{5}{4 x^{3}} + \frac{3 x^{2}}{2} + \frac{5 (- x^{4})}{10}$

Step 1.5.6.2

Cancel the common factors.

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Step 1.5.6.2.1

Factor $5$ out of $10$ .

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

Step 1.5.6.2.2

Cancel the common factor.

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

Step 1.5.6.2.3

Rewrite the expression.

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

Step 1.5.7

Move the negative in front of the fraction.

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

Step 1.6

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $4$ by $- 1$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Cancel the common factor of $- 4$ and $2$ .

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Step 2.2.6.1

Factor $2$ out of $- 4 x^{3}$ .

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

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

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

Step 2.2.6.2.2

Cancel the common factor.

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

Step 2.2.6.2.3

Rewrite the expression.

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

Step 2.2.6.2.4

Divide $- 2 x^{3}$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $2$ by $3$ .

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

Step 2.3.5

Combine $\frac{6}{2}$ and $x$ .

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

Step 2.3.6

Cancel the common factor of $6$ and $2$ .

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Step 2.3.6.1

Factor $2$ out of $6 x$ .

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

Step 2.3.6.2

Cancel the common factors.

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Step 2.3.6.2.1

Factor $2$ out of $2$ .

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

Step 2.3.6.2.2

Cancel the common factor.

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

Step 2.3.6.2.3

Rewrite the expression.

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

Step 2.3.6.2.4

Divide $3 x$ by $1$ .

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

Step 2.4

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

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Step 2.4.1

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5}{4 x^{3}}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

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

Step 2.4.2

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

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

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 $x^{3}$ .

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

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 $n = - 1$ .

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

Step 2.4.3.3

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

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

Step 2.4.4

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

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

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

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

Step 2.4.5.2

Multiply $3$ by $- 2$ .

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

Step 2.4.6

Multiply $3$ by $- 1$ .

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

Step 2.4.7

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

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Step 2.4.7.1

Move $x^{2}$ .

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

Step 2.4.7.2

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

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

Step 2.4.7.3

Subtract $6$ from $2$ .

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

Step 2.4.8

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

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

Step 2.4.9

Multiply $- 3$ by $5$ .

$- 2 x^{3} + 3 x + \frac{- 15}{4} x^{- 4} + \frac{d}{d x} [- 3]$

Step 2.4.10

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

$- 2 x^{3} + 3 x + \frac{- 15 x^{- 4}}{4} + \frac{d}{d x} [- 3]$

Step 2.4.11

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

$- 2 x^{3} + 3 x + \frac{- 15}{4 x^{4}} + \frac{d}{d x} [- 3]$

Step 2.4.12

Move the negative in front of the fraction.

$- 2 x^{3} + 3 x - \frac{15}{4 x^{4}} + \frac{d}{d x} [- 3]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 2 x^{3} + 3 x - \frac{15}{4 x^{4}} + 0$

Step 2.5.2

Add $- 2 x^{3} + 3 x - \frac{15}{4 x^{4}}$ and $0$ .

$f'' (x) = - 2 x^{3} + 3 x - \frac{15}{4 x^{4}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- 2 x^{3} + 3 x - \frac{15}{4 x^{4}}$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$ .

$\frac{d}{d x} [- 2 x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.2

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

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Step 3.2.1

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

$- 2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.2.2

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

$- 2 (3 x^{2}) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.2.3

Multiply $3$ by $- 2$ .

$- 6 x^{2} + \frac{d}{d x} [3 x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.3

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

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Step 3.3.1

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

$- 6 x^{2} + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.3.2

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

$- 6 x^{2} + 3 \cdot 1 + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.3.3

Multiply $3$ by $1$ .

$- 6 x^{2} + 3 + \frac{d}{d x} [- \frac{15}{4 x^{4}}]$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{15}{4 x^{4}}]$ .

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Step 3.4.1

Since $- \frac{15}{4}$ is constant with respect to $x$ , the derivative of $- \frac{15}{4 x^{4}}$ with respect to $x$ is $- \frac{15}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 6 x^{2} + 3 - \frac{15}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.4.2

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

$- 6 x^{2} + 3 - \frac{15}{4} \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 3.4.3

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

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Step 3.4.3.1

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

$- 6 x^{2} + 3 - \frac{15}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}])$

Step 3.4.3.2

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

$- 6 x^{2} + 3 - \frac{15}{4} (- u^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.4.3.3

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

$- 6 x^{2} + 3 - \frac{15}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.4.4

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

$- 6 x^{2} + 3 - \frac{15}{4} (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 3.4.5

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

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Step 3.4.5.1

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

$- 6 x^{2} + 3 - \frac{15}{4} (- x^{4 \cdot - 2} (4 x^{3}))$

Step 3.4.5.2

Multiply $4$ by $- 2$ .

$- 6 x^{2} + 3 - \frac{15}{4} (- x^{- 8} (4 x^{3}))$

Step 3.4.6

Multiply $4$ by $- 1$ .

$- 6 x^{2} + 3 - \frac{15}{4} (- 4 x^{- 8} x^{3})$

Step 3.4.7

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

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Step 3.4.7.1

Move $x^{3}$ .

$- 6 x^{2} + 3 - \frac{15}{4} (- 4 (x^{3} x^{- 8}))$

Step 3.4.7.2

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

$- 6 x^{2} + 3 - \frac{15}{4} (- 4 x^{3 - 8})$

Step 3.4.7.3

Subtract $8$ from $3$ .

$- 6 x^{2} + 3 - \frac{15}{4} (- 4 x^{- 5})$

Step 3.4.8

Multiply $- 4$ by $- 1$ .

$- 6 x^{2} + 3 + 4 (\frac{15}{4}) x^{- 5}$

Step 3.4.9

Combine $4$ and $\frac{15}{4}$ .

$- 6 x^{2} + 3 + \frac{4 \cdot 15}{4} x^{- 5}$

Step 3.4.10

Multiply $4$ by $15$ .

$- 6 x^{2} + 3 + \frac{60}{4} x^{- 5}$

Step 3.4.11

Combine $\frac{60}{4}$ and $x^{- 5}$ .

$- 6 x^{2} + 3 + \frac{60 x^{- 5}}{4}$

Step 3.4.12

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

$- 6 x^{2} + 3 + \frac{60}{4 x^{5}}$

Step 3.4.13

Cancel the common factor of $60$ and $4$ .

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Step 3.4.13.1

Factor $4$ out of $60$ .

$- 6 x^{2} + 3 + \frac{4 \cdot 15}{4 x^{5}}$

Step 3.4.13.2

Cancel the common factors.

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Step 3.4.13.2.1

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

$- 6 x^{2} + 3 + \frac{4 \cdot 15}{4 (x^{5})}$

Step 3.4.13.2.2

Cancel the common factor.

$- 6 x^{2} + 3 + \frac{4 \cdot 15}{4 x^{5}}$

Step 3.4.13.2.3

Rewrite the expression.

$- 6 x^{2} + 3 + \frac{15}{x^{5}}$

Step 3.5

Reorder terms.

$f''' (x) = - 6 x^{2} + \frac{15}{x^{5}} + 3$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 6 x^{2} + \frac{15}{x^{5}} + 3$ .

$- 6 x^{2} + \frac{15}{x^{5}} + 3$