Calculus Examples

$f (x) = \frac{5}{8} x^{- 2} + \frac{4}{3} x^{3} + \frac{2}{3} x^{4} + \frac{1}{10} x^{- 3}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{5}{8} x^{- 2} + \frac{4}{3} x^{3} + \frac{2}{3} x^{4} + \frac{1}{10} x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{8} x^{- 2}] + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$ .

$\frac{d}{d x} [\frac{5}{8} x^{- 2}] + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2

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

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Step 1.2.1

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

$\frac{5}{8} \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.2

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

$\frac{5}{8} (- 2 x^{- 3}) + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.3

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

$\frac{- 2 \cdot 5}{8} x^{- 3} + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.4

Multiply $- 2$ by $5$ .

$\frac{- 10}{8} x^{- 3} + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.5

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

$\frac{- 10 x^{- 3}}{8} + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.6

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

$\frac{- 10}{8 x^{3}} + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.7

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

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Step 1.2.7.1

Factor $2$ out of $- 10$ .

$\frac{2 (- 5)}{8 x^{3}} + \frac{d}{d x} [\frac{4}{3} x^{3}] + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

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

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

Step 1.2.7.2.2

Cancel the common factor.

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

Step 1.2.7.2.3

Rewrite the expression.

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

Step 1.2.8

Move the negative in front of the fraction.

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $3$ by $4$ .

$- \frac{5}{4 x^{3}} + \frac{12}{3} x^{2} + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.3.5

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

$- \frac{5}{4 x^{3}} + \frac{12 x^{2}}{3} + \frac{d}{d x} [\frac{2}{3} x^{4}] + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.3.6

Cancel the common factor of $12$ and $3$ .

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Step 1.3.6.1

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

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

Step 1.3.6.2

Cancel the common factors.

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Step 1.3.6.2.1

Factor $3$ out of $3$ .

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

Step 1.3.6.2.2

Cancel the common factor.

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

Step 1.3.6.2.3

Rewrite the expression.

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

Step 1.3.6.2.4

Divide $4 x^{2}$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $4$ by $2$ .

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8}{3} x^{3} + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.4.5

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

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{d}{d x} [\frac{1}{10} x^{- 3}]$

Step 1.5

Evaluate $\frac{d}{d x} [\frac{1}{10} x^{- 3}]$ .

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Step 1.5.1

Since $\frac{1}{10}$ is constant with respect to $x$ , the derivative of $\frac{1}{10} x^{- 3}$ with respect to $x$ is $\frac{1}{10} \frac{d}{d x} [x^{- 3}]$ .

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{1}{10} \frac{d}{d x} [x^{- 3}]$

Step 1.5.2

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

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{1}{10} (- 3 x^{- 4})$

Step 1.5.3

Combine $- 3$ and $\frac{1}{10}$ .

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{- 3}{10} x^{- 4}$

Step 1.5.4

Combine $\frac{- 3}{10}$ and $x^{- 4}$ .

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{- 3 x^{- 4}}{10}$

Step 1.5.5

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

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} + \frac{- 3}{10 x^{4}}$

Step 1.5.6

Move the negative in front of the fraction.

$- \frac{5}{4 x^{3}} + 4 x^{2} + \frac{8 x^{3}}{3} - \frac{3}{10 x^{4}}$

Step 1.6

Reorder terms.

$f' (x) = \frac{8 x^{3}}{3} + 4 x^{2} - \frac{5}{4 x^{3}} - \frac{3}{10 x^{4}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{8 x^{3}}{3} + 4 x^{2} - \frac{5}{4 x^{3}} - \frac{3}{10 x^{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{8 x^{3}}{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$ .

$\frac{d}{d x} [\frac{8 x^{3}}{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2

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

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Step 2.2.1

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

$\frac{8}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.2

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

$\frac{8}{3} (3 x^{2}) + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.3

Combine $3$ and $\frac{8}{3}$ .

$\frac{3 \cdot 8}{3} x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.4

Multiply $3$ by $8$ .

$\frac{24}{3} x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.5

Combine $\frac{24}{3}$ and $x^{2}$ .

$\frac{24 x^{2}}{3} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.6

Cancel the common factor of $24$ and $3$ .

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Step 2.2.6.1

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

$\frac{3 (8 x^{2})}{3} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (8 x^{2})}{3 (1)} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{3 (8 x^{2})}{3 \cdot 1} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{8 x^{2}}{1} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.2.6.2.4

Divide $8 x^{2}$ by $1$ .

$8 x^{2} + \frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.3

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

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Step 2.3.1

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

$8 x^{2} + 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.3.2

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

$8 x^{2} + 4 (2 x) + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.3.3

Multiply $2$ by $4$ .

$8 x^{2} + 8 x + \frac{d}{d x} [- \frac{5}{4 x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

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

$8 x^{2} + 8 x - \frac{5}{4} \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.2

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

$8 x^{2} + 8 x - \frac{5}{4} \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

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 $u_{1}$ as $x^{3}$ .

$8 x^{2} + 8 x - \frac{5}{4} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.3.2

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

$8 x^{2} + 8 x - \frac{5}{4} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.3.3

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

$8 x^{2} + 8 x - \frac{5}{4} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.4

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

$8 x^{2} + 8 x - \frac{5}{4} (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

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

$8 x^{2} + 8 x - \frac{5}{4} (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.5.2

Multiply $3$ by $- 2$ .

$8 x^{2} + 8 x - \frac{5}{4} (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.6

Multiply $3$ by $- 1$ .

$8 x^{2} + 8 x - \frac{5}{4} (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.7

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

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Step 2.4.7.1

Move $x^{2}$ .

$8 x^{2} + 8 x - \frac{5}{4} (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.7.2

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

$8 x^{2} + 8 x - \frac{5}{4} (- 3 x^{2 - 6}) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.7.3

Subtract $6$ from $2$ .

$8 x^{2} + 8 x - \frac{5}{4} (- 3 x^{- 4}) + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.8

Multiply $- 3$ by $- 1$ .

$8 x^{2} + 8 x + 3 (\frac{5}{4}) x^{- 4} + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.9

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

$8 x^{2} + 8 x + \frac{3 \cdot 5}{4} x^{- 4} + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.10

Multiply $3$ by $5$ .

$8 x^{2} + 8 x + \frac{15}{4} x^{- 4} + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.11

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

$8 x^{2} + 8 x + \frac{15 x^{- 4}}{4} + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.4.12

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{d}{d x} [- \frac{3}{10 x^{4}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- \frac{3}{10 x^{4}}]$ .

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Step 2.5.1

Since $- \frac{3}{10}$ is constant with respect to $x$ , the derivative of $- \frac{3}{10 x^{4}}$ with respect to $x$ is $- \frac{3}{10} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 2.5.2

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 2.5.3

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 2.5.3.1

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{4}])$

Step 2.5.3.2

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.5.3.3

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.5.4

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 2.5.5

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

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Step 2.5.5.1

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- x^{4 \cdot - 2} (4 x^{3}))$

Step 2.5.5.2

Multiply $4$ by $- 2$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- x^{- 8} (4 x^{3}))$

Step 2.5.6

Multiply $4$ by $- 1$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- 4 x^{- 8} x^{3})$

Step 2.5.7

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

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Step 2.5.7.1

Move $x^{3}$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- 4 (x^{3} x^{- 8}))$

Step 2.5.7.2

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- 4 x^{3 - 8})$

Step 2.5.7.3

Subtract $8$ from $3$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} - \frac{3}{10} (- 4 x^{- 5})$

Step 2.5.8

Multiply $- 4$ by $- 1$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + 4 (\frac{3}{10}) x^{- 5}$

Step 2.5.9

Combine $4$ and $\frac{3}{10}$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{4 \cdot 3}{10} x^{- 5}$

Step 2.5.10

Multiply $4$ by $3$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{12}{10} x^{- 5}$

Step 2.5.11

Combine $\frac{12}{10}$ and $x^{- 5}$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{12 x^{- 5}}{10}$

Step 2.5.12

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

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{12}{10 x^{5}}$

Step 2.5.13

Cancel the common factor of $12$ and $10$ .

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Step 2.5.13.1

Factor $2$ out of $12$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{2 (6)}{10 x^{5}}$

Step 2.5.13.2

Cancel the common factors.

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Step 2.5.13.2.1

Factor $2$ out of $10 x^{5}$ .

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{2 (6)}{2 (5 x^{5})}$

Step 2.5.13.2.2

Cancel the common factor.

$8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{2 \cdot 6}{2 (5 x^{5})}$

Step 2.5.13.2.3

Rewrite the expression.

$f'' (x) = 8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{6}{5 x^{5}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $8 x^{2} + 8 x + \frac{15}{4 x^{4}} + \frac{6}{5 x^{5}}$ with respect to $x$ is $\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$ .

$\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.2

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

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Step 3.2.1

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

$8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.2.2

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

$8 (2 x) + \frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.2.3

Multiply $2$ by $8$ .

$16 x + \frac{d}{d x} [8 x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.3

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

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Step 3.3.1

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

$16 x + 8 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.3.2

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

$16 x + 8 \cdot 1 + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.3.3

Multiply $8$ by $1$ .

$16 x + 8 + \frac{d}{d x} [\frac{15}{4 x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

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 $\frac{15}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$16 x + 8 + \frac{15}{4} \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.2

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

$16 x + 8 + \frac{15}{4} \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

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 $u_{1}$ as $x^{4}$ .

$16 x + 8 + \frac{15}{4} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.3.2

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

$16 x + 8 + \frac{15}{4} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.3.3

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

$16 x + 8 + \frac{15}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.4

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

$16 x + 8 + \frac{15}{4} (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

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

$16 x + 8 + \frac{15}{4} (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.5.2

Multiply $4$ by $- 2$ .

$16 x + 8 + \frac{15}{4} (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.6

Multiply $4$ by $- 1$ .

$16 x + 8 + \frac{15}{4} (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.7

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

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Step 3.4.7.1

Move $x^{3}$ .

$16 x + 8 + \frac{15}{4} (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.7.2

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

$16 x + 8 + \frac{15}{4} (- 4 x^{3 - 8}) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.7.3

Subtract $8$ from $3$ .

$16 x + 8 + \frac{15}{4} (- 4 x^{- 5}) + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.8

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

$16 x + 8 + \frac{- 4 \cdot 15}{4} x^{- 5} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.9

Multiply $- 4$ by $15$ .

$16 x + 8 + \frac{- 60}{4} x^{- 5} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.10

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

$16 x + 8 + \frac{- 60 x^{- 5}}{4} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.11

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

$16 x + 8 + \frac{- 60}{4 x^{5}} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.12

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

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Step 3.4.12.1

Factor $4$ out of $- 60$ .

$16 x + 8 + \frac{4 \cdot - 15}{4 x^{5}} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.12.2

Cancel the common factors.

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Step 3.4.12.2.1

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

$16 x + 8 + \frac{4 \cdot - 15}{4 (x^{5})} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.12.2.2

Cancel the common factor.

$16 x + 8 + \frac{4 \cdot - 15}{4 x^{5}} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.12.2.3

Rewrite the expression.

$16 x + 8 + \frac{- 15}{x^{5}} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.4.13

Move the negative in front of the fraction.

$16 x + 8 - \frac{15}{x^{5}} + \frac{d}{d x} [\frac{6}{5 x^{5}}]$

Step 3.5

Evaluate $\frac{d}{d x} [\frac{6}{5 x^{5}}]$ .

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Step 3.5.1

Since $\frac{6}{5}$ is constant with respect to $x$ , the derivative of $\frac{6}{5 x^{5}}$ with respect to $x$ is $\frac{6}{5} \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 3.5.2

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

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} \frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 3.5.3

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 3.5.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{5}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{5}])$

Step 3.5.3.2

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

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{5}])$

Step 3.5.3.3

Replace all occurrences of $u_{2}$ with $x^{5}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}])$

Step 3.5.4

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

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- {(x^{5})}^{- 2} (5 x^{4}))$

Step 3.5.5

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

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Step 3.5.5.1

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

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- x^{5 \cdot - 2} (5 x^{4}))$

Step 3.5.5.2

Multiply $5$ by $- 2$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- x^{- 10} (5 x^{4}))$

Step 3.5.6

Multiply $5$ by $- 1$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- 5 x^{- 10} x^{4})$

Step 3.5.7

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

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Step 3.5.7.1

Move $x^{4}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- 5 (x^{4} x^{- 10}))$

Step 3.5.7.2

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

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- 5 x^{4 - 10})$

Step 3.5.7.3

Subtract $10$ from $4$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{6}{5} (- 5 x^{- 6})$

Step 3.5.8

Combine $- 5$ and $\frac{6}{5}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{- 5 \cdot 6}{5} x^{- 6}$

Step 3.5.9

Multiply $- 5$ by $6$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{- 30}{5} x^{- 6}$

Step 3.5.10

Combine $\frac{- 30}{5}$ and $x^{- 6}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{- 30 x^{- 6}}{5}$

Step 3.5.11

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{- 30}{5 x^{6}}$

Step 3.5.12

Cancel the common factor of $- 30$ and $5$ .

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Step 3.5.12.1

Factor $5$ out of $- 30$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{5 \cdot - 6}{5 x^{6}}$

Step 3.5.12.2

Cancel the common factors.

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Step 3.5.12.2.1

Factor $5$ out of $5 x^{6}$ .

$16 x + 8 - \frac{15}{x^{5}} + \frac{5 \cdot - 6}{5 (x^{6})}$

Step 3.5.12.2.2

Cancel the common factor.

$16 x + 8 - \frac{15}{x^{5}} + \frac{5 \cdot - 6}{5 x^{6}}$

Step 3.5.12.2.3

Rewrite the expression.

$16 x + 8 - \frac{15}{x^{5}} + \frac{- 6}{x^{6}}$

Step 3.5.13

Move the negative in front of the fraction.

$16 x + 8 - \frac{15}{x^{5}} - \frac{6}{x^{6}}$

Step 3.6

Reorder terms.

$f''' (x) = 16 x - \frac{15}{x^{5}} - \frac{6}{x^{6}} + 8$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $16 x - \frac{15}{x^{5}} - \frac{6}{x^{6}} + 8$ .

$16 x - \frac{15}{x^{5}} - \frac{6}{x^{6}} + 8$