Calculus Examples

$f (x) = \frac{1}{24} x^{6} + \frac{1}{15} x^{- 4} - \frac{1}{5} x^{5} + 2 x^{- 1}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{24} x^{6} + \frac{1}{15} x^{- 4} - \frac{1}{5} x^{5} + 2 x^{- 1}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{24} x^{6}] + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$ .

$\frac{d}{d x} [\frac{1}{24} x^{6}] + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{24} x^{6}]$ .

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Step 1.2.1

Since $\frac{1}{24}$ is constant with respect to $x$ , the derivative of $\frac{1}{24} x^{6}$ with respect to $x$ is $\frac{1}{24} \frac{d}{d x} [x^{6}]$ .

$\frac{1}{24} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.2

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

$\frac{1}{24} (6 x^{5}) + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.3

Combine $6$ and $\frac{1}{24}$ .

$\frac{6}{24} x^{5} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.4

Combine $\frac{6}{24}$ and $x^{5}$ .

$\frac{6 x^{5}}{24} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.5

Cancel the common factor of $6$ and $24$ .

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Step 1.2.5.1

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

$\frac{6 (x^{5})}{24} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.5.2

Cancel the common factors.

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Step 1.2.5.2.1

Factor $6$ out of $24$ .

$\frac{6 x^{5}}{6 \cdot 4} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.5.2.2

Cancel the common factor.

$\frac{6 x^{5}}{6 \cdot 4} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.2.5.2.3

Rewrite the expression.

$\frac{x^{5}}{4} + \frac{d}{d x} [\frac{1}{15} x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{x^{5}}{4} + \frac{1}{15} \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.3.2

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

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

Step 1.3.3

Combine $- 4$ and $\frac{1}{15}$ .

$\frac{x^{5}}{4} + \frac{- 4}{15} x^{- 5} + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.3.4

Combine $\frac{- 4}{15}$ and $x^{- 5}$ .

$\frac{x^{5}}{4} + \frac{- 4 x^{- 5}}{15} + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.3.5

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

$\frac{x^{5}}{4} + \frac{- 4}{15 x^{5}} + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.3.6

Move the negative in front of the fraction.

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} + \frac{d}{d x} [- \frac{1}{5} x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4

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

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Step 1.4.1

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - \frac{1}{5} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4.2

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

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

Step 1.4.3

Multiply $5$ by $- 1$ .

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

Step 1.4.4

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} + \frac{- 5}{5} x^{4} + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4.5

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} + \frac{- 5 x^{4}}{5} + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $5$ out of $5$ .

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

Step 1.4.6.2.2

Cancel the common factor.

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} + \frac{5 (- x^{4})}{5 \cdot 1} + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4.6.2.3

Rewrite the expression.

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} + \frac{- x^{4}}{1} + \frac{d}{d x} [2 x^{- 1}]$

Step 1.4.6.2.4

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - x^{4} + \frac{d}{d x} [2 x^{- 1}]$

Step 1.5

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

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Step 1.5.1

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - x^{4} + 2 \frac{d}{d x} [x^{- 1}]$

Step 1.5.2

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

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

Step 1.5.3

Multiply $- 1$ by $2$ .

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - x^{4} - 2 x^{- 2}$

Step 1.6

Simplify.

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Step 1.6.1

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

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - x^{4} - 2 \frac{1}{x^{2}}$

Step 1.6.2

Combine terms.

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Step 1.6.2.1

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

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

Step 1.6.2.2

Move the negative in front of the fraction.

$\frac{x^{5}}{4} - \frac{4}{15 x^{5}} - x^{4} - \frac{2}{x^{2}}$

Step 1.6.3

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{4} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- x^{4}] + \frac{d}{d x} [- \frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.2.2

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

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

Step 2.2.3

Combine $5$ and $\frac{1}{4}$ .

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

Step 2.2.4

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $4$ by $- 1$ .

$\frac{5 x^{4}}{4} - 4 x^{3} + \frac{d}{d x} [- \frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.4

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

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Step 2.4.1

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

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.4.2

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

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

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

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Step 2.4.3.1

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

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

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 $n = - 1$ .

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.4.3.3

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

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

Step 2.4.4

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

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

Multiply $2$ by $- 2$ .

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

Step 2.4.6

Multiply $2$ by $- 1$ .

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

Step 2.4.7

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

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.4.8

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

$\frac{5 x^{4}}{4} - 4 x^{3} - 2 (- 2 x^{1 - 4}) + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.4.9

Subtract $4$ from $1$ .

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

Step 2.4.10

Multiply $- 2$ by $- 2$ .

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} + \frac{d}{d x} [- \frac{4}{15 x^{5}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- \frac{4}{15 x^{5}}]$ .

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Step 2.5.1

Since $- \frac{4}{15}$ is constant with respect to $x$ , the derivative of $- \frac{4}{15 x^{5}}$ with respect to $x$ is $- \frac{4}{15} \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} - \frac{4}{15} \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.5.2

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

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} - \frac{4}{15} \frac{d}{d x} [{(x^{5})}^{- 1}]$

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

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Step 2.5.3.1

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

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} - \frac{4}{15} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{5}])$

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 $n = - 1$ .

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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

Step 2.5.5

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

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Step 2.5.5.1

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

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

Step 2.5.5.2

Multiply $5$ by $- 2$ .

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

Step 2.5.6

Multiply $5$ by $- 1$ .

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

Step 2.5.7

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

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Step 2.5.7.1

Move $x^{4}$ .

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

Step 2.5.7.2

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

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

Step 2.5.7.3

Subtract $10$ from $4$ .

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

Step 2.5.8

Multiply $- 5$ by $- 1$ .

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

Step 2.5.9

Combine $5$ and $\frac{4}{15}$ .

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

Step 2.5.10

Multiply $5$ by $4$ .

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} + \frac{20}{15} x^{- 6}$

Step 2.5.11

Combine $\frac{20}{15}$ and $x^{- 6}$ .

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} + \frac{20 x^{- 6}}{15}$

Step 2.5.12

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

$\frac{5 x^{4}}{4} - 4 x^{3} + 4 x^{- 3} + \frac{20}{15 x^{6}}$

Step 2.5.13

Cancel the common factor of $20$ and $15$ .

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Step 2.5.13.1

Factor $5$ out of $20$ .

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

Step 2.5.13.2

Cancel the common factors.

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Step 2.5.13.2.1

Factor $5$ out of $15 x^{6}$ .

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

Step 2.5.13.2.2

Cancel the common factor.

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

Step 2.5.13.2.3

Rewrite the expression.

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

Step 2.6

Simplify.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $4$ by $5$ .

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

Step 3.2.5

Combine $\frac{20}{4}$ and $x^{3}$ .

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

Step 3.2.6

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

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Step 3.2.6.1

Factor $4$ out of $20 x^{3}$ .

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

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

Factor $4$ out of $4$ .

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

Step 3.2.6.2.2

Cancel the common factor.

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

Step 3.2.6.2.3

Rewrite the expression.

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

Step 3.2.6.2.4

Divide $5 x^{3}$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply $3$ by $- 4$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

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

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Step 3.4.3.1

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

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

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 $n = - 1$ .

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

Step 3.4.3.3

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

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

Step 3.4.4

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

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

Step 3.4.5

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

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Step 3.4.5.1

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

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

Step 3.4.5.2

Multiply $3$ by $- 2$ .

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

Step 3.4.6

Multiply $3$ by $- 1$ .

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

Step 3.4.7

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

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Step 3.4.7.1

Move $x^{2}$ .

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

Step 3.4.7.2

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

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

Step 3.4.7.3

Subtract $6$ from $2$ .

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

Step 3.4.8

Multiply $- 3$ by $4$ .

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

Step 3.5

Evaluate $\frac{d}{d x} [\frac{4}{3 x^{6}}]$ .

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Step 3.5.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4}{3 x^{6}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{1}{x^{6}}]$ .

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

Step 3.5.2

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

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

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

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Step 3.5.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{6}$ .

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

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 $n = - 1$ .

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

Step 3.5.3.3

Replace all occurrences of $u_{2}$ with $x^{6}$ .

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

Step 3.5.4

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

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

Step 3.5.5

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

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Step 3.5.5.1

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

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

Step 3.5.5.2

Multiply $6$ by $- 2$ .

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

Step 3.5.6

Multiply $6$ by $- 1$ .

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

Step 3.5.7

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

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Step 3.5.7.1

Move $x^{5}$ .

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

Step 3.5.7.2

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

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

Step 3.5.7.3

Subtract $12$ from $5$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{4}{3} (- 6 x^{- 7})$

Step 3.5.8

Combine $- 6$ and $\frac{4}{3}$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{- 6 \cdot 4}{3} x^{- 7}$

Step 3.5.9

Multiply $- 6$ by $4$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{- 24}{3} x^{- 7}$

Step 3.5.10

Combine $\frac{- 24}{3}$ and $x^{- 7}$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{- 24 x^{- 7}}{3}$

Step 3.5.11

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

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{- 24}{3 x^{7}}$

Step 3.5.12

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

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Step 3.5.12.1

Factor $3$ out of $- 24$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{3 \cdot - 8}{3 x^{7}}$

Step 3.5.12.2

Cancel the common factors.

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Step 3.5.12.2.1

Factor $3$ out of $3 x^{7}$ .

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{3 \cdot - 8}{3 (x^{7})}$

Step 3.5.12.2.2

Cancel the common factor.

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{3 \cdot - 8}{3 x^{7}}$

Step 3.5.12.2.3

Rewrite the expression.

$5 x^{3} - 12 x^{2} - 12 x^{- 4} + \frac{- 8}{x^{7}}$

Step 3.5.13

Move the negative in front of the fraction.

$5 x^{3} - 12 x^{2} - 12 x^{- 4} - \frac{8}{x^{7}}$

Step 3.6

Simplify.

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Step 3.6.1

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

$5 x^{3} - 12 x^{2} - 12 \frac{1}{x^{4}} - \frac{8}{x^{7}}$

Step 3.6.2

Combine terms.

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Step 3.6.2.1

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

$5 x^{3} - 12 x^{2} + \frac{- 12}{x^{4}} - \frac{8}{x^{7}}$

Step 3.6.2.2

Move the negative in front of the fraction.

$f''' (x) = 5 x^{3} - 12 x^{2} - \frac{12}{x^{4}} - \frac{8}{x^{7}}$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $5 x^{3} - 12 x^{2} - \frac{12}{x^{4}} - \frac{8}{x^{7}}$ .

$5 x^{3} - 12 x^{2} - \frac{12}{x^{4}} - \frac{8}{x^{7}}$