Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{5}$ .

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

Step 1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.2.5

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

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

Step 1.2.6

Combine the numerators over the common denominator.

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

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $5$ .

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

Step 1.2.7.2

Subtract $5$ from $3$ .

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

Step 1.2.8

Move the negative in front of the fraction.

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

Step 1.2.9

Combine $\frac{3}{5}$ and $x^{- \frac{2}{5}}$ .

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

Step 1.2.10

Combine $4$ and $\frac{3 x^{- \frac{2}{5}}}{5}$ .

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

Step 1.2.11

Multiply $3$ by $4$ .

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

Step 1.2.12

Move $x^{- \frac{2}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

Multiply $2$ by $- 2$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{1}{8} (- x^{- 4} (2 x)) + \frac{d}{d x} [- \sqrt{x}]$

Step 1.3.6

Multiply $2$ by $- 1$ .

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{1}{8} (- 2 x^{- 4} x) + \frac{d}{d x} [- \sqrt{x}]$

Step 1.3.7

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

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{1}{8} (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [- \sqrt{x}]$

Step 1.3.8

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

$\frac{12}{5 x^{\frac{2}{5}}} - \frac{1}{8} (- 2 x^{1 - 4}) + \frac{d}{d x} [- \sqrt{x}]$

Step 1.3.9

Subtract $4$ from $1$ .

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

Step 1.3.10

Multiply $- 2$ by $- 1$ .

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

Step 1.3.11

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

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

Step 1.3.12

Combine $\frac{2}{8}$ and $x^{- 3}$ .

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

Step 1.3.13

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

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

Step 1.3.14

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

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Step 1.3.14.1

Factor $2$ out of $2$ .

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

Step 1.3.14.2

Cancel the common factors.

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Step 1.3.14.2.1

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

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

Step 1.3.14.2.2

Cancel the common factor.

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

Step 1.3.14.2.3

Rewrite the expression.

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

Step 1.4

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

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.4.5

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

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

Step 1.4.6

Combine the numerators over the common denominator.

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

Step 1.4.7

Simplify the numerator.

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Step 1.4.7.1

Multiply $- 1$ by $2$ .

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

Step 1.4.7.2

Subtract $2$ from $1$ .

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

Step 1.4.8

Move the negative in front of the fraction.

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

Step 1.4.9

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

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

Step 1.4.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.5

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{1}{4 x^{3}} + \frac{12}{5 x^{\frac{2}{5}}} - \frac{1}{2 x^{\frac{1}{2}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4 x^{3}}] + \frac{d}{d x} [\frac{12}{5 x^{\frac{2}{5}}}] + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $3$ by $- 2$ .

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

Step 2.2.6

Multiply $3$ by $- 1$ .

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

Step 2.2.7

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

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Step 2.2.7.1

Move $x^{2}$ .

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

Step 2.2.7.2

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

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

Step 2.2.7.3

Subtract $6$ from $2$ .

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

Step 2.2.8

Combine $- 3$ and $\frac{1}{4}$ .

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

Step 2.2.9

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

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

Step 2.2.10

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

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

Step 2.2.11

Move the negative in front of the fraction.

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

Step 2.3

Evaluate $\frac{d}{d x} [\frac{12}{5 x^{\frac{2}{5}}}]$ .

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Step 2.3.1

Since $\frac{12}{5}$ is constant with respect to $x$ , the derivative of $\frac{12}{5 x^{\frac{2}{5}}}$ with respect to $x$ is $\frac{12}{5} \frac{d}{d x} [\frac{1}{x^{\frac{2}{5}}}]$ .

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

Step 2.3.2

Rewrite $\frac{1}{x^{\frac{2}{5}}}$ as ${(x^{\frac{2}{5}})}^{- 1}$ .

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

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{2}{5}}$ .

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Step 2.3.3.1

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

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

Step 2.3.3.2

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

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

Step 2.3.3.3

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

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

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{5}$ .

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

Step 2.3.5

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

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Step 2.3.5.1

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

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

Step 2.3.5.2

Multiply $\frac{2}{5} \cdot - 2$ .

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

Multiply $2$ by $- 2$ .

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

Step 2.3.5.3

Move the negative in front of the fraction.

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

Step 2.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.3.7

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

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

Step 2.3.8

Combine the numerators over the common denominator.

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

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $5$ .

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

Step 2.3.9.2

Subtract $5$ from $2$ .

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

Step 2.3.10

Move the negative in front of the fraction.

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

Step 2.3.11

Combine $\frac{2}{5}$ and $x^{- \frac{3}{5}}$ .

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

Step 2.3.12

Combine $\frac{2 x^{- \frac{3}{5}}}{5}$ and $x^{- \frac{4}{5}}$ .

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

Step 2.3.13

Multiply $x^{- \frac{3}{5}}$ by $x^{- \frac{4}{5}}$ by adding the exponents.

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Step 2.3.13.1

Move $x^{- \frac{4}{5}}$ .

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

Step 2.3.13.2

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

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

Step 2.3.13.3

Combine the numerators over the common denominator.

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

Step 2.3.13.4

Subtract $3$ from $- 4$ .

$- \frac{3}{4 x^{4}} + \frac{12}{5} (- \frac{2 x^{\frac{- 7}{5}}}{5}) + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.3.13.5

Move the negative in front of the fraction.

$- \frac{3}{4 x^{4}} + \frac{12}{5} (- \frac{2 x^{- \frac{7}{5}}}{5}) + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.3.14

Move $x^{- \frac{7}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{3}{4 x^{4}} + \frac{12}{5} (- \frac{2}{5 x^{\frac{7}{5}}}) + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.3.15

Multiply $\frac{12}{5}$ by $\frac{2}{5 x^{\frac{7}{5}}}$ .

$- \frac{3}{4 x^{4}} - \frac{12 \cdot 2}{5 (5 x^{\frac{7}{5}})} + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.3.16

Multiply $12$ by $2$ .

$- \frac{3}{4 x^{4}} - \frac{24}{5 (5 x^{\frac{7}{5}})} + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.3.17

Multiply $5$ by $5$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} + \frac{d}{d x} [- \frac{1}{2 x^{\frac{1}{2}}}]$

Step 2.4

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

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Step 2.4.1

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$

Step 2.4.2

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}]$

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{1}{2}}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.4.3.2

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.4.3.3

Replace all occurrences of $u_{3}$ with $x^{\frac{1}{2}}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 2.4.4

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.4.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 2.4.5.1

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.4.5.2

Cancel the common factor of $2$ .

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Step 2.4.5.2.1

Factor $2$ out of $- 2$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.4.5.2.2

Cancel the common factor.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.4.5.2.3

Rewrite the expression.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1}))$

Step 2.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 2.4.7

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 2.4.8

Combine the numerators over the common denominator.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}))$

Step 2.4.9

Simplify the numerator.

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Step 2.4.9.1

Multiply $- 1$ by $2$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}}))$

Step 2.4.9.2

Subtract $2$ from $1$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}}))$

Step 2.4.10

Move the negative in front of the fraction.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}}))$

Step 2.4.11

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2})$

Step 2.4.12

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2})$

Step 2.4.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.4.13.1

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{- \frac{1}{2} - 1}}{2})$

Step 2.4.13.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 2.4.13.3

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

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 2.4.13.4

Combine the numerators over the common denominator.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 2.4.13.5

Simplify the numerator.

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Step 2.4.13.5.1

Multiply $- 1$ by $2$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{\frac{- 1 - 2}{2}}}{2})$

Step 2.4.13.5.2

Subtract $2$ from $- 1$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{\frac{- 3}{2}}}{2})$

Step 2.4.13.6

Move the negative in front of the fraction.

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{x^{- \frac{3}{2}}}{2})$

Step 2.4.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} - \frac{1}{2} (- \frac{1}{2 x^{\frac{3}{2}}})$

Step 2.4.15

Multiply $- 1$ by $- 1$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} + 1 (\frac{1}{2}) \frac{1}{2 x^{\frac{3}{2}}}$

Step 2.4.16

Multiply $\frac{1}{2}$ by $1$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} + \frac{1}{2} \cdot \frac{1}{2 x^{\frac{3}{2}}}$

Step 2.4.17

Multiply $\frac{1}{2}$ by $\frac{1}{2 x^{\frac{3}{2}}}$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} + \frac{1}{2 (2 x^{\frac{3}{2}})}$

Step 2.4.18

Multiply $2$ by $2$ .

$- \frac{3}{4 x^{4}} - \frac{24}{25 x^{\frac{7}{5}}} + \frac{1}{4 x^{\frac{3}{2}}}$

Step 2.5

Reorder terms.

$f'' (x) = - \frac{3}{4 x^{4}} + \frac{1}{4 x^{\frac{3}{2}}} - \frac{24}{25 x^{\frac{7}{5}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{3}{4 x^{4}} + \frac{1}{4 x^{\frac{3}{2}}} - \frac{24}{25 x^{\frac{7}{5}}}$ with respect to $x$ is $\frac{d}{d x} [- \frac{3}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$ .

$\frac{d}{d x} [- \frac{3}{4 x^{4}}] + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2

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

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Step 3.2.1

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

$- \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.2

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

$- \frac{3}{4} \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{4}$ .

$- \frac{3}{4} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.3.2

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

$- \frac{3}{4} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.3.3

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

$- \frac{3}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.4

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

$- \frac{3}{4} (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.5

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

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Step 3.2.5.1

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

$- \frac{3}{4} (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.5.2

Multiply $4$ by $- 2$ .

$- \frac{3}{4} (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.6

Multiply $4$ by $- 1$ .

$- \frac{3}{4} (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.7

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

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Step 3.2.7.1

Move $x^{3}$ .

$- \frac{3}{4} (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.7.2

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

$- \frac{3}{4} (- 4 x^{3 - 8}) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.7.3

Subtract $8$ from $3$ .

$- \frac{3}{4} (- 4 x^{- 5}) + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.8

Multiply $- 4$ by $- 1$ .

$4 (\frac{3}{4}) x^{- 5} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.9

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

$\frac{4 \cdot 3}{4} x^{- 5} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.10

Multiply $4$ by $3$ .

$\frac{12}{4} x^{- 5} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.11

Combine $\frac{12}{4}$ and $x^{- 5}$ .

$\frac{12 x^{- 5}}{4} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.12

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

$\frac{12}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.13

Cancel the common factor of $12$ and $4$ .

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Step 3.2.13.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 3}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.13.2

Cancel the common factors.

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Step 3.2.13.2.1

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

$\frac{4 \cdot 3}{4 (x^{5})} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.13.2.2

Cancel the common factor.

$\frac{4 \cdot 3}{4 x^{5}} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.2.13.2.3

Rewrite the expression.

$\frac{3}{x^{5}} + \frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{1}{4 x^{\frac{3}{2}}}]$ .

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Step 3.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4 x^{\frac{3}{2}}}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}]$ .

$\frac{3}{x^{5}} + \frac{1}{4} \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$\frac{3}{x^{5}} + \frac{1}{4} \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}] + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

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 $g (x) = x^{\frac{3}{2}}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{3}{2}}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.3.2

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

$\frac{3}{x^{5}} + \frac{1}{4} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{3}{2}}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- {(x^{\frac{3}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.5

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

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Step 3.3.5.1

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

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.5.2

Cancel the common factor of $2$ .

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Step 3.3.5.2.1

Factor $2$ out of $- 2$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.5.2.2

Cancel the common factor.

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.5.2.3

Rewrite the expression.

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.5.3

Multiply $3$ by $- 1$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.7

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

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.8

Combine the numerators over the common denominator.

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $2$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.9.2

Subtract $2$ from $3$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} (\frac{3}{2} x^{\frac{1}{2}})) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.10

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

$\frac{3}{x^{5}} + \frac{1}{4} (- x^{- 3} \frac{3 x^{\frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.11

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $x^{- 3}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{\frac{1}{2}} x^{- 3}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12

Multiply $x^{\frac{1}{2}}$ by $x^{- 3}$ by adding the exponents.

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Step 3.3.12.1

Move $x^{- 3}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.2

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

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{- 3 + \frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.4

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

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.5

Combine the numerators over the common denominator.

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{\frac{- 3 \cdot 2 + 1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.6

Simplify the numerator.

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Step 3.3.12.6.1

Multiply $- 3$ by $2$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{\frac{- 6 + 1}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.6.2

Add $- 6$ and $1$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{\frac{- 5}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.12.7

Move the negative in front of the fraction.

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3 x^{- \frac{5}{2}}}{2}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.13

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{x^{5}} + \frac{1}{4} (- \frac{3}{2 x^{\frac{5}{2}}}) + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.14

Multiply $\frac{1}{4}$ by $\frac{3}{2 x^{\frac{5}{2}}}$ .

$\frac{3}{x^{5}} - \frac{3}{4 (2 x^{\frac{5}{2}})} + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.3.15

Multiply $2$ by $4$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + \frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$

Step 3.4

Evaluate $\frac{d}{d x} [- \frac{24}{25 x^{\frac{7}{5}}}]$ .

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Step 3.4.1

Since $- \frac{24}{25}$ is constant with respect to $x$ , the derivative of $- \frac{24}{25 x^{\frac{7}{5}}}$ with respect to $x$ is $- \frac{24}{25} \frac{d}{d x} [\frac{1}{x^{\frac{7}{5}}}]$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} \frac{d}{d x} [\frac{1}{x^{\frac{7}{5}}}]$

Step 3.4.2

Rewrite $\frac{1}{x^{\frac{7}{5}}}$ as ${(x^{\frac{7}{5}})}^{- 1}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} \frac{d}{d x} [{(x^{\frac{7}{5}})}^{- 1}]$

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 $g (x) = x^{\frac{7}{5}}$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{7}{5}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{7}{5}}])$

Step 3.4.3.2

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

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{7}{5}}])$

Step 3.4.3.3

Replace all occurrences of $u_{3}$ with $x^{\frac{7}{5}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- {(x^{\frac{7}{5}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{5}}])$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{5}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- {(x^{\frac{7}{5}})}^{- 2} (\frac{7}{5} x^{\frac{7}{5} - 1}))$

Step 3.4.5

Multiply the exponents in ${(x^{\frac{7}{5}})}^{- 2}$ .

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Step 3.4.5.1

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

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{\frac{7}{5} \cdot - 2} (\frac{7}{5} x^{\frac{7}{5} - 1}))$

Step 3.4.5.2

Multiply $\frac{7}{5} \cdot - 2$ .

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Step 3.4.5.2.1

Combine $\frac{7}{5}$ and $- 2$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{\frac{7 \cdot - 2}{5}} (\frac{7}{5} x^{\frac{7}{5} - 1}))$

Step 3.4.5.2.2

Multiply $7$ by $- 2$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{\frac{- 14}{5}} (\frac{7}{5} x^{\frac{7}{5} - 1}))$

Step 3.4.5.3

Move the negative in front of the fraction.

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{7}{5} - 1}))$

Step 3.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{7}{5} - 1 \cdot \frac{5}{5}}))$

Step 3.4.7

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

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{7}{5} + \frac{- 1 \cdot 5}{5}}))$

Step 3.4.8

Combine the numerators over the common denominator.

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{7 - 1 \cdot 5}{5}}))$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $5$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{7 - 5}{5}}))$

Step 3.4.9.2

Subtract $5$ from $7$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} (\frac{7}{5} x^{\frac{2}{5}}))$

Step 3.4.10

Combine $\frac{7}{5}$ and $x^{\frac{2}{5}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- x^{- \frac{14}{5}} \frac{7 x^{\frac{2}{5}}}{5})$

Step 3.4.11

Combine $\frac{7 x^{\frac{2}{5}}}{5}$ and $x^{- \frac{14}{5}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 x^{\frac{2}{5}} x^{- \frac{14}{5}}}{5})$

Step 3.4.12

Multiply $x^{\frac{2}{5}}$ by $x^{- \frac{14}{5}}$ by adding the exponents.

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Step 3.4.12.1

Move $x^{- \frac{14}{5}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 (x^{- \frac{14}{5}} x^{\frac{2}{5}})}{5})$

Step 3.4.12.2

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

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 x^{- \frac{14}{5} + \frac{2}{5}}}{5})$

Step 3.4.12.3

Combine the numerators over the common denominator.

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 x^{\frac{- 14 + 2}{5}}}{5})$

Step 3.4.12.4

Add $- 14$ and $2$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 x^{\frac{- 12}{5}}}{5})$

Step 3.4.12.5

Move the negative in front of the fraction.

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7 x^{- \frac{12}{5}}}{5})$

Step 3.4.13

Move $x^{- \frac{12}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} - \frac{24}{25} (- \frac{7}{5 x^{\frac{12}{5}}})$

Step 3.4.14

Multiply $- 1$ by $- 1$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + 1 (\frac{24}{25}) \frac{7}{5 x^{\frac{12}{5}}}$

Step 3.4.15

Multiply $\frac{24}{25}$ by $1$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + \frac{24}{25} \cdot \frac{7}{5 x^{\frac{12}{5}}}$

Step 3.4.16

Multiply $\frac{24}{25}$ by $\frac{7}{5 x^{\frac{12}{5}}}$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + \frac{24 \cdot 7}{25 (5 x^{\frac{12}{5}})}$

Step 3.4.17

Multiply $24$ by $7$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + \frac{168}{25 (5 x^{\frac{12}{5}})}$

Step 3.4.18

Multiply $5$ by $25$ .

$\frac{3}{x^{5}} - \frac{3}{8 x^{\frac{5}{2}}} + \frac{168}{125 x^{\frac{12}{5}}}$

Step 3.5

Reorder terms.

$f''' (x) = \frac{3}{x^{5}} + \frac{168}{125 x^{\frac{12}{5}}} - \frac{3}{8 x^{\frac{5}{2}}}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $\frac{3}{x^{5}} + \frac{168}{125 x^{\frac{12}{5}}} - \frac{3}{8 x^{\frac{5}{2}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{x^{5}}] + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$ .

$\frac{d}{d x} [\frac{3}{x^{5}}] + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2

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

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Step 4.2.1

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

$3 \frac{d}{d x} [\frac{1}{x^{5}}] + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.2

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

$3 \frac{d}{d x} [{(x^{5})}^{- 1}] + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{5}$ .

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Step 4.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{5}$ .

$3 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.3.2

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

$3 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.3.3

Replace all occurrences of $u_{1}$ with $x^{5}$ .

$3 (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.4

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

$3 (- {(x^{5})}^{- 2} (5 x^{4})) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.5

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

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Step 4.2.5.1

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

$3 (- x^{5 \cdot - 2} (5 x^{4})) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.5.2

Multiply $5$ by $- 2$ .

$3 (- x^{- 10} (5 x^{4})) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.6

Multiply $5$ by $- 1$ .

$3 (- 5 x^{- 10} x^{4}) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.7

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

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Step 4.2.7.1

Move $x^{4}$ .

$3 (- 5 (x^{4} x^{- 10})) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.7.2

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

$3 (- 5 x^{4 - 10}) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.7.3

Subtract $10$ from $4$ .

$3 (- 5 x^{- 6}) + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.2.8

Multiply $- 5$ by $3$ .

$- 15 x^{- 6} + \frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3

Evaluate $\frac{d}{d x} [\frac{168}{125 x^{\frac{12}{5}}}]$ .

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Step 4.3.1

Since $\frac{168}{125}$ is constant with respect to $x$ , the derivative of $\frac{168}{125 x^{\frac{12}{5}}}$ with respect to $x$ is $\frac{168}{125} \frac{d}{d x} [\frac{1}{x^{\frac{12}{5}}}]$ .

$- 15 x^{- 6} + \frac{168}{125} \frac{d}{d x} [\frac{1}{x^{\frac{12}{5}}}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.2

Rewrite $\frac{1}{x^{\frac{12}{5}}}$ as ${(x^{\frac{12}{5}})}^{- 1}$ .

$- 15 x^{- 6} + \frac{168}{125} \frac{d}{d x} [{(x^{\frac{12}{5}})}^{- 1}] + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{12}{5}}$ .

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Step 4.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{12}{5}}$ .

$- 15 x^{- 6} + \frac{168}{125} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{12}{5}}]) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.3.2

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

$- 15 x^{- 6} + \frac{168}{125} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{12}{5}}]) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{12}{5}}$ .

$- 15 x^{- 6} + \frac{168}{125} (- {(x^{\frac{12}{5}})}^{- 2} \frac{d}{d x} [x^{\frac{12}{5}}]) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{12}{5}$ .

$- 15 x^{- 6} + \frac{168}{125} (- {(x^{\frac{12}{5}})}^{- 2} (\frac{12}{5} x^{\frac{12}{5} - 1})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.5

Multiply the exponents in ${(x^{\frac{12}{5}})}^{- 2}$ .

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Step 4.3.5.1

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

$- 15 x^{- 6} + \frac{168}{125} (- x^{\frac{12}{5} \cdot - 2} (\frac{12}{5} x^{\frac{12}{5} - 1})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.5.2

Multiply $\frac{12}{5} \cdot - 2$ .

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Step 4.3.5.2.1

Combine $\frac{12}{5}$ and $- 2$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{\frac{12 \cdot - 2}{5}} (\frac{12}{5} x^{\frac{12}{5} - 1})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.5.2.2

Multiply $12$ by $- 2$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{\frac{- 24}{5}} (\frac{12}{5} x^{\frac{12}{5} - 1})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.5.3

Move the negative in front of the fraction.

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{12}{5} - 1})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{12}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.7

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

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{12}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.8

Combine the numerators over the common denominator.

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{12 - 1 \cdot 5}{5}})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.9

Simplify the numerator.

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Step 4.3.9.1

Multiply $- 1$ by $5$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{12 - 5}{5}})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.9.2

Subtract $5$ from $12$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} (\frac{12}{5} x^{\frac{7}{5}})) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.10

Combine $\frac{12}{5}$ and $x^{\frac{7}{5}}$ .

$- 15 x^{- 6} + \frac{168}{125} (- x^{- \frac{24}{5}} \frac{12 x^{\frac{7}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.11

Combine $\frac{12 x^{\frac{7}{5}}}{5}$ and $x^{- \frac{24}{5}}$ .

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 x^{\frac{7}{5}} x^{- \frac{24}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.12

Multiply $x^{\frac{7}{5}}$ by $x^{- \frac{24}{5}}$ by adding the exponents.

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Step 4.3.12.1

Move $x^{- \frac{24}{5}}$ .

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 (x^{- \frac{24}{5}} x^{\frac{7}{5}})}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.12.2

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

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 x^{- \frac{24}{5} + \frac{7}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.12.3

Combine the numerators over the common denominator.

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 x^{\frac{- 24 + 7}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.12.4

Add $- 24$ and $7$ .

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 x^{\frac{- 17}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.12.5

Move the negative in front of the fraction.

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12 x^{- \frac{17}{5}}}{5}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.13

Move $x^{- \frac{17}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 15 x^{- 6} + \frac{168}{125} (- \frac{12}{5 x^{\frac{17}{5}}}) + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.14

Multiply $\frac{168}{125}$ by $\frac{12}{5 x^{\frac{17}{5}}}$ .

$- 15 x^{- 6} - \frac{168 \cdot 12}{125 (5 x^{\frac{17}{5}})} + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.15

Multiply $168$ by $12$ .

$- 15 x^{- 6} - \frac{2016}{125 (5 x^{\frac{17}{5}})} + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.3.16

Multiply $5$ by $125$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$

Step 4.4

Evaluate $\frac{d}{d x} [- \frac{3}{8 x^{\frac{5}{2}}}]$ .

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Step 4.4.1

Since $- \frac{3}{8}$ is constant with respect to $x$ , the derivative of $- \frac{3}{8 x^{\frac{5}{2}}}$ with respect to $x$ is $- \frac{3}{8} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} \frac{d}{d x} [\frac{1}{x^{\frac{5}{2}}}]$

Step 4.4.2

Rewrite $\frac{1}{x^{\frac{5}{2}}}$ as ${(x^{\frac{5}{2}})}^{- 1}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} \frac{d}{d x} [{(x^{\frac{5}{2}})}^{- 1}]$

Step 4.4.3

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 $g (x) = x^{\frac{5}{2}}$ .

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Step 4.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{5}{2}}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.4.3.2

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

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.4.3.3

Replace all occurrences of $u_{3}$ with $x^{\frac{5}{2}}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- {(x^{\frac{5}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{2}}])$

Step 4.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- {(x^{\frac{5}{2}})}^{- 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.5

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

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Step 4.4.5.1

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

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{\frac{5}{2} \cdot - 2} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.5.2

Cancel the common factor of $2$ .

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Step 4.4.5.2.1

Factor $2$ out of $- 2$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{\frac{5}{2} \cdot (2 (- 1))} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.5.2.2

Cancel the common factor.

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{\frac{5}{2} \cdot (2 \cdot - 1)} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.5.2.3

Rewrite the expression.

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{5 \cdot - 1} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.5.3

Multiply $5$ by $- 1$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1}))$

Step 4.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}}))$

Step 4.4.7

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

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 4.4.8

Combine the numerators over the common denominator.

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}}))$

Step 4.4.9

Simplify the numerator.

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Step 4.4.9.1

Multiply $- 1$ by $2$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{5 - 2}{2}}))$

Step 4.4.9.2

Subtract $2$ from $5$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} (\frac{5}{2} x^{\frac{3}{2}}))$

Step 4.4.10

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- x^{- 5} \frac{5 x^{\frac{3}{2}}}{2})$

Step 4.4.11

Combine $\frac{5 x^{\frac{3}{2}}}{2}$ and $x^{- 5}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{\frac{3}{2}} x^{- 5}}{2})$

Step 4.4.12

Multiply $x^{\frac{3}{2}}$ by $x^{- 5}$ by adding the exponents.

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Step 4.4.12.1

Move $x^{- 5}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 (x^{- 5} x^{\frac{3}{2}})}{2})$

Step 4.4.12.2

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

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{- 5 + \frac{3}{2}}}{2})$

Step 4.4.12.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{- 5 \cdot \frac{2}{2} + \frac{3}{2}}}{2})$

Step 4.4.12.4

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

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{\frac{- 5 \cdot 2}{2} + \frac{3}{2}}}{2})$

Step 4.4.12.5

Combine the numerators over the common denominator.

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{\frac{- 5 \cdot 2 + 3}{2}}}{2})$

Step 4.4.12.6

Simplify the numerator.

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Step 4.4.12.6.1

Multiply $- 5$ by $2$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{\frac{- 10 + 3}{2}}}{2})$

Step 4.4.12.6.2

Add $- 10$ and $3$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{\frac{- 7}{2}}}{2})$

Step 4.4.12.7

Move the negative in front of the fraction.

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5 x^{- \frac{7}{2}}}{2})$

Step 4.4.13

Move $x^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} - \frac{3}{8} (- \frac{5}{2 x^{\frac{7}{2}}})$

Step 4.4.14

Multiply $- 1$ by $- 1$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + 1 (\frac{3}{8}) \frac{5}{2 x^{\frac{7}{2}}}$

Step 4.4.15

Multiply $\frac{3}{8}$ by $1$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{3}{8} \cdot \frac{5}{2 x^{\frac{7}{2}}}$

Step 4.4.16

Multiply $\frac{3}{8}$ by $\frac{5}{2 x^{\frac{7}{2}}}$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{3 \cdot 5}{8 (2 x^{\frac{7}{2}})}$

Step 4.4.17

Multiply $3$ by $5$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{15}{8 (2 x^{\frac{7}{2}})}$

Step 4.4.18

Multiply $2$ by $8$ .

$- 15 x^{- 6} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{15}{16 x^{\frac{7}{2}}}$

Step 4.5

Simplify.

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Step 4.5.1

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

$- 15 \frac{1}{x^{6}} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{15}{16 x^{\frac{7}{2}}}$

Step 4.5.2

Combine terms.

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Step 4.5.2.1

Combine $- 15$ and $\frac{1}{x^{6}}$ .

$\frac{- 15}{x^{6}} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{15}{16 x^{\frac{7}{2}}}$

Step 4.5.2.2

Move the negative in front of the fraction.

$- \frac{15}{x^{6}} - \frac{2016}{625 x^{\frac{17}{5}}} + \frac{15}{16 x^{\frac{7}{2}}}$

Step 4.5.3

Reorder terms.

$f^{4} (x) = - \frac{15}{x^{6}} + \frac{15}{16 x^{\frac{7}{2}}} - \frac{2016}{625 x^{\frac{17}{5}}}$