Calculus Examples

$y = 6 \sqrt[3]{x^{2}} - 20 \sqrt[5]{x^{4}} + \frac{7}{\sqrt[7]{x^{5}}} + \frac{4}{\sqrt[6]{x}}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $6 \sqrt[3]{x^{2}} - 20 \sqrt[5]{x^{4}} + \frac{7}{\sqrt[7]{x^{5}}} + \frac{4}{\sqrt[6]{x}}$ with respect to $x$ is $\frac{d}{d x} [6 \sqrt[3]{x^{2}}] + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$ .

$\frac{d}{d x} [6 \sqrt[3]{x^{2}}] + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2

Evaluate $\frac{d}{d x} [6 \sqrt[3]{x^{2}}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [6 x^{\frac{2}{3}}] + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{\frac{2}{3}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$6 \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.3

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

$6 (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$6 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.5

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

$6 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.6

Combine the numerators over the common denominator.

$6 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $3$ .

$6 (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.7.2

Subtract $3$ from $2$ .

$6 (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.8

Move the negative in front of the fraction.

$6 (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.9

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

$6 \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.10

Combine $6$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{6 (2 x^{- \frac{1}{3}})}{3} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.11

Multiply $2$ by $6$ .

$\frac{12 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.12

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

$\frac{12}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.13

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.14

Cancel the common factors.

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Step 1.2.14.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$\frac{3 \cdot 4}{3 (x^{\frac{1}{3}})} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.14.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.2.14.3

Rewrite the expression.

$\frac{4}{x^{\frac{1}{3}}} + \frac{d}{d x} [- 20 \sqrt[5]{x^{4}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3

Evaluate $\frac{d}{d x} [- 20 \sqrt[5]{x^{4}}]$ .

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{4}}$ as $x^{\frac{4}{5}}$ .

$\frac{4}{x^{\frac{1}{3}}} + \frac{d}{d x} [- 20 x^{\frac{4}{5}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.2

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x^{\frac{4}{5}}$ with respect to $x$ is $- 20 \frac{d}{d x} [x^{\frac{4}{5}}]$ .

$\frac{4}{x^{\frac{1}{3}}} - 20 \frac{d}{d x} [x^{\frac{4}{5}}] + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{5}$ .

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{4}{5} - 1}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.4

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

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{4}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.5

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

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.6

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{4 - 1 \cdot 5}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.7

Simplify the numerator.

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Step 1.3.7.1

Multiply $- 1$ by $5$ .

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{4 - 5}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.7.2

Subtract $5$ from $4$ .

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{\frac{- 1}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.8

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - 20 (\frac{4}{5} x^{- \frac{1}{5}}) + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.9

Combine $\frac{4}{5}$ and $x^{- \frac{1}{5}}$ .

$\frac{4}{x^{\frac{1}{3}}} - 20 \frac{4 x^{- \frac{1}{5}}}{5} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.10

Combine $- 20$ and $\frac{4 x^{- \frac{1}{5}}}{5}$ .

$\frac{4}{x^{\frac{1}{3}}} + \frac{- 20 (4 x^{- \frac{1}{5}})}{5} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.11

Multiply $4$ by $- 20$ .

$\frac{4}{x^{\frac{1}{3}}} + \frac{- 80 x^{- \frac{1}{5}}}{5} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.12

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

$\frac{4}{x^{\frac{1}{3}}} + \frac{- 80}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.13

Factor $5$ out of $- 80$ .

$\frac{4}{x^{\frac{1}{3}}} + \frac{5 \cdot - 16}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.14

Cancel the common factors.

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Step 1.3.14.1

Factor $5$ out of $5 x^{\frac{1}{5}}$ .

$\frac{4}{x^{\frac{1}{3}}} + \frac{5 \cdot - 16}{5 (x^{\frac{1}{5}})} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.14.2

Cancel the common factor.

$\frac{4}{x^{\frac{1}{3}}} + \frac{5 \cdot - 16}{5 x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.14.3

Rewrite the expression.

$\frac{4}{x^{\frac{1}{3}}} + \frac{- 16}{x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.3.15

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{7}{\sqrt[7]{x^{5}}}]$ .

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{5}}$ as $x^{\frac{5}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{d}{d x} [\frac{7}{x^{\frac{5}{7}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.2

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 \frac{d}{d x} [\frac{1}{x^{\frac{5}{7}}}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.3

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 \frac{d}{d x} [{(x^{\frac{5}{7}})}^{- 1}] + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.4

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

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Step 1.4.4.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{5}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{5}{7}}]) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.4.2

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{5}{7}}]) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.4.3

Replace all occurrences of $u_{1}$ with $x^{\frac{5}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- {(x^{\frac{5}{7}})}^{- 2} \frac{d}{d x} [x^{\frac{5}{7}}]) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.5

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- {(x^{\frac{5}{7}})}^{- 2} (\frac{5}{7} x^{\frac{5}{7} - 1})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.6

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

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Step 1.4.6.1

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{\frac{5}{7} \cdot - 2} (\frac{5}{7} x^{\frac{5}{7} - 1})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.6.2

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

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Step 1.4.6.2.1

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{\frac{5 \cdot - 2}{7}} (\frac{5}{7} x^{\frac{5}{7} - 1})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.6.2.2

Multiply $5$ by $- 2$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{\frac{- 10}{7}} (\frac{5}{7} x^{\frac{5}{7} - 1})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.6.3

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{5}{7} - 1})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{5}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.8

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{5}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.9

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{5 - 1 \cdot 7}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.10

Simplify the numerator.

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Step 1.4.10.1

Multiply $- 1$ by $7$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{5 - 7}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.10.2

Subtract $7$ from $5$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{\frac{- 2}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.11

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} (\frac{5}{7} x^{- \frac{2}{7}})) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.12

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- x^{- \frac{10}{7}} \frac{5 x^{- \frac{2}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.13

Combine $\frac{5 x^{- \frac{2}{7}}}{7}$ and $x^{- \frac{10}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 x^{- \frac{2}{7}} x^{- \frac{10}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.14

Multiply $x^{- \frac{2}{7}}$ by $x^{- \frac{10}{7}}$ by adding the exponents.

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Step 1.4.14.1

Move $x^{- \frac{10}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 (x^{- \frac{10}{7}} x^{- \frac{2}{7}})}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.14.2

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 x^{- \frac{10}{7} - \frac{2}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.14.3

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 x^{\frac{- 10 - 2}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.14.4

Subtract $2$ from $- 10$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 x^{\frac{- 12}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.14.5

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5 x^{- \frac{12}{7}}}{7}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.15

Move $x^{- \frac{12}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + 7 (- \frac{5}{7 x^{\frac{12}{7}}}) + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.16

Multiply $- 1$ by $7$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - 7 \frac{5}{7 x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.17

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{- 7 \cdot 5}{7 x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.18

Multiply $- 7$ by $5$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{- 35}{7 x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.19

Factor $7$ out of $- 35$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{7 \cdot - 5}{7 x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.20

Cancel the common factors.

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Step 1.4.20.1

Factor $7$ out of $7 x^{\frac{12}{7}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{7 \cdot - 5}{7 (x^{\frac{12}{7}})} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.20.2

Cancel the common factor.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{7 \cdot - 5}{7 x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.20.3

Rewrite the expression.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} + \frac{- 5}{x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.4.21

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{\sqrt[6]{x}}]$

Step 1.5

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

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Step 1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{d}{d x} [\frac{4}{x^{\frac{1}{6}}}]$

Step 1.5.2

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{6}}}]$

Step 1.5.3

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 \frac{d}{d x} [{(x^{\frac{1}{6}})}^{- 1}]$

Step 1.5.4

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 $g (x) = x^{\frac{1}{6}}$ .

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Step 1.5.4.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{6}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{6}}])$

Step 1.5.4.2

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{1}{6}}])$

Step 1.5.4.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{6}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- {(x^{\frac{1}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{6}}])$

Step 1.5.5

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- {(x^{\frac{1}{6}})}^{- 2} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6

Multiply the exponents in ${(x^{\frac{1}{6}})}^{- 2}$ .

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Step 1.5.6.1

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{1}{6} \cdot - 2} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.2

Cancel the common factor of $2$ .

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Step 1.5.6.2.1

Factor $2$ out of $6$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{1}{2 (3)} \cdot - 2} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.2.2

Factor $2$ out of $- 2$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{1}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.2.3

Cancel the common factor.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{1}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.2.4

Rewrite the expression.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{1}{3} \cdot - 1} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.3

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{\frac{- 1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.6.4

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1}))$

Step 1.5.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} - 1 \cdot \frac{6}{6}}))$

Step 1.5.8

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1}{6} + \frac{- 1 \cdot 6}{6}}))$

Step 1.5.9

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1 - 1 \cdot 6}{6}}))$

Step 1.5.10

Simplify the numerator.

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Step 1.5.10.1

Multiply $- 1$ by $6$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{1 - 6}{6}}))$

Step 1.5.10.2

Subtract $6$ from $1$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{\frac{- 5}{6}}))$

Step 1.5.11

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} (\frac{1}{6} x^{- \frac{5}{6}}))$

Step 1.5.12

Combine $\frac{1}{6}$ and $x^{- \frac{5}{6}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- x^{- \frac{1}{3}} \frac{x^{- \frac{5}{6}}}{6})$

Step 1.5.13

Combine $\frac{x^{- \frac{5}{6}}}{6}$ and $x^{- \frac{1}{3}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{5}{6}} x^{- \frac{1}{3}}}{6})$

Step 1.5.14

Multiply $x^{- \frac{5}{6}}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 1.5.14.1

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{5}{6} - \frac{1}{3}}}{6})$

Step 1.5.14.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{5}{6} - \frac{1}{3} \cdot \frac{2}{2}}}{6})$

Step 1.5.14.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.14.3.1

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

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{5}{6} - \frac{2}{3 \cdot 2}}}{6})$

Step 1.5.14.3.2

Multiply $3$ by $2$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{5}{6} - \frac{2}{6}}}{6})$

Step 1.5.14.4

Combine the numerators over the common denominator.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{\frac{- 5 - 2}{6}}}{6})$

Step 1.5.14.5

Subtract $2$ from $- 5$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{\frac{- 7}{6}}}{6})$

Step 1.5.14.6

Move the negative in front of the fraction.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{x^{- \frac{7}{6}}}{6})$

Step 1.5.15

Move $x^{- \frac{7}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + 4 (- \frac{1}{6 x^{\frac{7}{6}}})$

Step 1.5.16

Multiply $- 1$ by $4$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} - 4 \frac{1}{6 x^{\frac{7}{6}}}$

Step 1.5.17

Combine $- 4$ and $\frac{1}{6 x^{\frac{7}{6}}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{- 4}{6 x^{\frac{7}{6}}}$

Step 1.5.18

Factor $2$ out of $- 4$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{2 (- 2)}{6 x^{\frac{7}{6}}}$

Step 1.5.19

Cancel the common factors.

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Step 1.5.19.1

Factor $2$ out of $6 x^{\frac{7}{6}}$ .

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{2 (- 2)}{2 (3 x^{\frac{7}{6}})}$

Step 1.5.19.2

Cancel the common factor.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{2 \cdot - 2}{2 (3 x^{\frac{7}{6}})}$

Step 1.5.19.3

Rewrite the expression.

$\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} + \frac{- 2}{3 x^{\frac{7}{6}}}$

Step 1.5.20

Move the negative in front of the fraction.

$f' (x) = \frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} - \frac{2}{3 x^{\frac{7}{6}}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{4}{x^{\frac{1}{3}}} - \frac{16}{x^{\frac{1}{5}}} - \frac{5}{x^{\frac{12}{7}}} - \frac{2}{3 x^{\frac{7}{6}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{4}{x^{\frac{1}{3}}}] + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$ .

$\frac{d}{d x} [\frac{4}{x^{\frac{1}{3}}}] + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2

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

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Step 2.2.1

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

$4 \frac{d}{d x} [\frac{1}{x^{\frac{1}{3}}}] + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.2

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

$4 \frac{d}{d x} [{(x^{\frac{1}{3}})}^{- 1}] + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $g (x) = x^{\frac{1}{3}}$ .

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Step 2.2.3.1

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

$4 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $n = - 1$ .

$4 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.3.3

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

$4 (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{3}}]) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.4

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

$4 (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 2}$ .

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Step 2.2.5.1

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

$4 (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.5.2

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

$4 (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.5.3

Move the negative in front of the fraction.

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.7

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

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.8

Combine the numerators over the common denominator.

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $- 1$ by $3$ .

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.9.2

Subtract $3$ from $1$ .

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.10

Move the negative in front of the fraction.

$4 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}})) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.11

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

$4 (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.12

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

$4 (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.13

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

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Step 2.2.13.1

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

$4 (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.13.2

Combine the numerators over the common denominator.

$4 (- \frac{x^{\frac{- 2 - 2}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.13.3

Subtract $2$ from $- 2$ .

$4 (- \frac{x^{\frac{- 4}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.13.4

Move the negative in front of the fraction.

$4 (- \frac{x^{- \frac{4}{3}}}{3}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.14

Move $x^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{3 x^{\frac{4}{3}}}) + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.15

Multiply $- 1$ by $4$ .

$- 4 \frac{1}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.16

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

$\frac{- 4}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.2.17

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{16}{x^{\frac{1}{5}}}]$ .

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Step 2.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16}{x^{\frac{1}{5}}}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{1}{x^{\frac{1}{5}}}]$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 \frac{d}{d x} [\frac{1}{x^{\frac{1}{5}}}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.2

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 \frac{d}{d x} [{(x^{\frac{1}{5}})}^{- 1}] + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $g (x) = x^{\frac{1}{5}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{5}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{5}}]) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $n = - 1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{1}{5}}]) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{5}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- {(x^{\frac{1}{5}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{5}}]) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.4

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- {(x^{\frac{1}{5}})}^{- 2} (\frac{1}{5} x^{\frac{1}{5} - 1})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.5

Multiply the exponents in ${(x^{\frac{1}{5}})}^{- 2}$ .

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Step 2.3.5.1

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{\frac{1}{5} \cdot - 2} (\frac{1}{5} x^{\frac{1}{5} - 1})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.5.2

Combine $\frac{1}{5}$ and $- 2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{\frac{- 2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.5.3

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.6

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.7

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.8

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $5$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{1 - 5}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.9.2

Subtract $5$ from $1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{\frac{- 4}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.10

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} (\frac{1}{5} x^{- \frac{4}{5}})) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.11

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- x^{- \frac{2}{5}} \frac{x^{- \frac{4}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.12

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{x^{- \frac{4}{5}} x^{- \frac{2}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.13

Multiply $x^{- \frac{4}{5}}$ by $x^{- \frac{2}{5}}$ by adding the exponents.

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Step 2.3.13.1

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

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{x^{- \frac{4}{5} - \frac{2}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.13.2

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{x^{\frac{- 4 - 2}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.13.3

Subtract $2$ from $- 4$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{x^{\frac{- 6}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.13.4

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{x^{- \frac{6}{5}}}{5}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.14

Move $x^{- \frac{6}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} - 16 (- \frac{1}{5 x^{\frac{6}{5}}}) + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.15

Multiply $- 1$ by $- 16$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + 16 \frac{1}{5 x^{\frac{6}{5}}} + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.3.16

Combine $16$ and $\frac{1}{5 x^{\frac{6}{5}}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4

Evaluate $\frac{d}{d x} [- \frac{5}{x^{\frac{12}{7}}}]$ .

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Step 2.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x^{\frac{12}{7}}}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x^{\frac{12}{7}}}]$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 \frac{d}{d x} [\frac{1}{x^{\frac{12}{7}}}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.2

Rewrite $\frac{1}{x^{\frac{12}{7}}}$ as ${(x^{\frac{12}{7}})}^{- 1}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 \frac{d}{d x} [{(x^{\frac{12}{7}})}^{- 1}] + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $g (x) = x^{\frac{12}{7}}$ .

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Step 2.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{12}{7}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{12}{7}}]) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

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 $n = - 1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{12}{7}}]) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.3.3

Replace all occurrences of $u_{3}$ with $x^{\frac{12}{7}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- {(x^{\frac{12}{7}})}^{- 2} \frac{d}{d x} [x^{\frac{12}{7}}]) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{12}{7}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- {(x^{\frac{12}{7}})}^{- 2} (\frac{12}{7} x^{\frac{12}{7} - 1})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.5

Multiply the exponents in ${(x^{\frac{12}{7}})}^{- 2}$ .

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Step 2.4.5.1

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{\frac{12}{7} \cdot - 2} (\frac{12}{7} x^{\frac{12}{7} - 1})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.5.2

Multiply $\frac{12}{7} \cdot - 2$ .

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Step 2.4.5.2.1

Combine $\frac{12}{7}$ and $- 2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{\frac{12 \cdot - 2}{7}} (\frac{12}{7} x^{\frac{12}{7} - 1})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.5.2.2

Multiply $12$ by $- 2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{\frac{- 24}{7}} (\frac{12}{7} x^{\frac{12}{7} - 1})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.5.3

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{12}{7} - 1})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{12}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.7

Combine $- 1$ and $\frac{7}{7}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{12}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.8

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{12 - 1 \cdot 7}{7}})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.9

Simplify the numerator.

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Step 2.4.9.1

Multiply $- 1$ by $7$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{12 - 7}{7}})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.9.2

Subtract $7$ from $12$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} (\frac{12}{7} x^{\frac{5}{7}})) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.10

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- x^{- \frac{24}{7}} \frac{12 x^{\frac{5}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.11

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 x^{\frac{5}{7}} x^{- \frac{24}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.12

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

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Step 2.4.12.1

Move $x^{- \frac{24}{7}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 (x^{- \frac{24}{7}} x^{\frac{5}{7}})}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.12.2

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 x^{- \frac{24}{7} + \frac{5}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.12.3

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 x^{\frac{- 24 + 5}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.12.4

Add $- 24$ and $5$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 x^{\frac{- 19}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.12.5

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12 x^{- \frac{19}{7}}}{7}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.13

Move $x^{- \frac{19}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} - 5 (- \frac{12}{7 x^{\frac{19}{7}}}) + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.14

Multiply $- 1$ by $- 5$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + 5 \frac{12}{7 x^{\frac{19}{7}}} + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.15

Combine $5$ and $\frac{12}{7 x^{\frac{19}{7}}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{5 \cdot 12}{7 x^{\frac{19}{7}}} + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.4.16

Multiply $5$ by $12$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- \frac{2}{3 x^{\frac{7}{6}}}]$ .

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Step 2.5.1

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3 x^{\frac{7}{6}}}$ with respect to $x$ is $- \frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{7}{6}}}]$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} \frac{d}{d x} [\frac{1}{x^{\frac{7}{6}}}]$

Step 2.5.2

Rewrite $\frac{1}{x^{\frac{7}{6}}}$ as ${(x^{\frac{7}{6}})}^{- 1}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} \frac{d}{d x} [{(x^{\frac{7}{6}})}^{- 1}]$

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 $g (x) = x^{\frac{7}{6}}$ .

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Step 2.5.3.1

To apply the Chain Rule, set $u_{4}$ as $x^{\frac{7}{6}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (\frac{d}{d u_{4}} [{u_{4}}^{- 1}] \frac{d}{d x} [x^{\frac{7}{6}}])$

Step 2.5.3.2

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- {u_{4}}^{- 2} \frac{d}{d x} [x^{\frac{7}{6}}])$

Step 2.5.3.3

Replace all occurrences of $u_{4}$ with $x^{\frac{7}{6}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- {(x^{\frac{7}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{7}{6}}])$

Step 2.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{6}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- {(x^{\frac{7}{6}})}^{- 2} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5

Multiply the exponents in ${(x^{\frac{7}{6}})}^{- 2}$ .

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Step 2.5.5.1

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7}{6} \cdot - 2} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.2

Cancel the common factor of $2$ .

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Step 2.5.5.2.1

Factor $2$ out of $6$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7}{2 (3)} \cdot - 2} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.2.2

Factor $2$ out of $- 2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.2.3

Cancel the common factor.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.2.4

Rewrite the expression.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7}{3} \cdot - 1} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.3

Combine $\frac{7}{3}$ and $- 1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{7 \cdot - 1}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.4

Multiply $7$ by $- 1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{\frac{- 7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.5.5

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1}))$

Step 2.5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} - 1 \cdot \frac{6}{6}}))$

Step 2.5.7

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7}{6} + \frac{- 1 \cdot 6}{6}}))$

Step 2.5.8

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7 - 1 \cdot 6}{6}}))$

Step 2.5.9

Simplify the numerator.

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Step 2.5.9.1

Multiply $- 1$ by $6$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{7 - 6}{6}}))$

Step 2.5.9.2

Subtract $6$ from $7$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} (\frac{7}{6} x^{\frac{1}{6}}))$

Step 2.5.10

Combine $\frac{7}{6}$ and $x^{\frac{1}{6}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- x^{- \frac{7}{3}} \frac{7 x^{\frac{1}{6}}}{6})$

Step 2.5.11

Combine $\frac{7 x^{\frac{1}{6}}}{6}$ and $x^{- \frac{7}{3}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{\frac{1}{6}} x^{- \frac{7}{3}}}{6})$

Step 2.5.12

Multiply $x^{\frac{1}{6}}$ by $x^{- \frac{7}{3}}$ by adding the exponents.

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Step 2.5.12.1

Move $x^{- \frac{7}{3}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 (x^{- \frac{7}{3}} x^{\frac{1}{6}})}{6})$

Step 2.5.12.2

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{- \frac{7}{3} + \frac{1}{6}}}{6})$

Step 2.5.12.3

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{- \frac{7}{3} \cdot \frac{2}{2} + \frac{1}{6}}}{6})$

Step 2.5.12.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.12.4.1

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{- \frac{7 \cdot 2}{3 \cdot 2} + \frac{1}{6}}}{6})$

Step 2.5.12.4.2

Multiply $3$ by $2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{- \frac{7 \cdot 2}{6} + \frac{1}{6}}}{6})$

Step 2.5.12.5

Combine the numerators over the common denominator.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{\frac{- 7 \cdot 2 + 1}{6}}}{6})$

Step 2.5.12.6

Simplify the numerator.

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Step 2.5.12.6.1

Multiply $- 7$ by $2$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{\frac{- 14 + 1}{6}}}{6})$

Step 2.5.12.6.2

Add $- 14$ and $1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{\frac{- 13}{6}}}{6})$

Step 2.5.12.7

Move the negative in front of the fraction.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7 x^{- \frac{13}{6}}}{6})$

Step 2.5.13

Move $x^{- \frac{13}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} - \frac{2}{3} (- \frac{7}{6 x^{\frac{13}{6}}})$

Step 2.5.14

Multiply $- 1$ by $- 1$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + 1 (\frac{2}{3}) \frac{7}{6 x^{\frac{13}{6}}}$

Step 2.5.15

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

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{2}{3} \cdot \frac{7}{6 x^{\frac{13}{6}}}$

Step 2.5.16

Multiply $\frac{2}{3}$ by $\frac{7}{6 x^{\frac{13}{6}}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{2 \cdot 7}{3 (6 x^{\frac{13}{6}})}$

Step 2.5.17

Multiply $2$ by $7$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{14}{3 (6 x^{\frac{13}{6}})}$

Step 2.5.18

Multiply $6$ by $3$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{14}{18 x^{\frac{13}{6}}}$

Step 2.5.19

Factor $2$ out of $14$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{2 (7)}{18 x^{\frac{13}{6}}}$

Step 2.5.20

Cancel the common factors.

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Step 2.5.20.1

Factor $2$ out of $18 x^{\frac{13}{6}}$ .

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{2 (7)}{2 (9 x^{\frac{13}{6}})}$

Step 2.5.20.2

Cancel the common factor.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{2 \cdot 7}{2 (9 x^{\frac{13}{6}})}$

Step 2.5.20.3

Rewrite the expression.

$- \frac{4}{3 x^{\frac{4}{3}}} + \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{7}{9 x^{\frac{13}{6}}}$

Step 2.6

Reorder terms.

$f'' (x) = \frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{7}{9 x^{\frac{13}{6}}} - \frac{4}{3 x^{\frac{4}{3}}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{16}{5 x^{\frac{6}{5}}} + \frac{60}{7 x^{\frac{19}{7}}} + \frac{7}{9 x^{\frac{13}{6}}} - \frac{4}{3 x^{\frac{4}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{16}{5 x^{\frac{6}{5}}}] + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$ .

$\frac{d}{d x} [\frac{16}{5 x^{\frac{6}{5}}}] + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{16}{5 x^{\frac{6}{5}}}]$ .

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Step 3.2.1

Since $\frac{16}{5}$ is constant with respect to $x$ , the derivative of $\frac{16}{5 x^{\frac{6}{5}}}$ with respect to $x$ is $\frac{16}{5} \frac{d}{d x} [\frac{1}{x^{\frac{6}{5}}}]$ .

$\frac{16}{5} \frac{d}{d x} [\frac{1}{x^{\frac{6}{5}}}] + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.2

Rewrite $\frac{1}{x^{\frac{6}{5}}}$ as ${(x^{\frac{6}{5}})}^{- 1}$ .

$\frac{16}{5} \frac{d}{d x} [{(x^{\frac{6}{5}})}^{- 1}] + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $g (x) = x^{\frac{6}{5}}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{6}{5}}$ .

$\frac{16}{5} (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $n = - 1$ .

$\frac{16}{5} (- {u_{1}}^{- 2} \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{6}{5}}$ .

$\frac{16}{5} (- {(x^{\frac{6}{5}})}^{- 2} \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{6}{5}$ .

$\frac{16}{5} (- {(x^{\frac{6}{5}})}^{- 2} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.5

Multiply the exponents in ${(x^{\frac{6}{5}})}^{- 2}$ .

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Step 3.2.5.1

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

$\frac{16}{5} (- x^{\frac{6}{5} \cdot - 2} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.5.2

Multiply $\frac{6}{5} \cdot - 2$ .

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Step 3.2.5.2.1

Combine $\frac{6}{5}$ and $- 2$ .

$\frac{16}{5} (- x^{\frac{6 \cdot - 2}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.5.2.2

Multiply $6$ by $- 2$ .

$\frac{16}{5} (- x^{\frac{- 12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.5.3

Move the negative in front of the fraction.

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.6

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

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.7

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

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.8

Combine the numerators over the common denominator.

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6 - 1 \cdot 5}{5}})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.9

Simplify the numerator.

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Step 3.2.9.1

Multiply $- 1$ by $5$ .

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6 - 5}{5}})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.9.2

Subtract $5$ from $6$ .

$\frac{16}{5} (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{1}{5}})) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.10

Combine $\frac{6}{5}$ and $x^{\frac{1}{5}}$ .

$\frac{16}{5} (- x^{- \frac{12}{5}} \frac{6 x^{\frac{1}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.11

Combine $\frac{6 x^{\frac{1}{5}}}{5}$ and $x^{- \frac{12}{5}}$ .

$\frac{16}{5} (- \frac{6 x^{\frac{1}{5}} x^{- \frac{12}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.12

Multiply $x^{\frac{1}{5}}$ by $x^{- \frac{12}{5}}$ by adding the exponents.

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Step 3.2.12.1

Move $x^{- \frac{12}{5}}$ .

$\frac{16}{5} (- \frac{6 (x^{- \frac{12}{5}} x^{\frac{1}{5}})}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.12.2

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

$\frac{16}{5} (- \frac{6 x^{- \frac{12}{5} + \frac{1}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.12.3

Combine the numerators over the common denominator.

$\frac{16}{5} (- \frac{6 x^{\frac{- 12 + 1}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.12.4

Add $- 12$ and $1$ .

$\frac{16}{5} (- \frac{6 x^{\frac{- 11}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.12.5

Move the negative in front of the fraction.

$\frac{16}{5} (- \frac{6 x^{- \frac{11}{5}}}{5}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.13

Move $x^{- \frac{11}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{16}{5} (- \frac{6}{5 x^{\frac{11}{5}}}) + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.14

Multiply $\frac{16}{5}$ by $\frac{6}{5 x^{\frac{11}{5}}}$ .

$- \frac{16 \cdot 6}{5 (5 x^{\frac{11}{5}})} + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.15

Multiply $16$ by $6$ .

$- \frac{96}{5 (5 x^{\frac{11}{5}})} + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.2.16

Multiply $5$ by $5$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{60}{7 x^{\frac{19}{7}}}]$ .

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Step 3.3.1

Since $\frac{60}{7}$ is constant with respect to $x$ , the derivative of $\frac{60}{7 x^{\frac{19}{7}}}$ with respect to $x$ is $\frac{60}{7} \frac{d}{d x} [\frac{1}{x^{\frac{19}{7}}}]$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} \frac{d}{d x} [\frac{1}{x^{\frac{19}{7}}}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{\frac{19}{7}}}$ as ${(x^{\frac{19}{7}})}^{- 1}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} \frac{d}{d x} [{(x^{\frac{19}{7}})}^{- 1}] + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $g (x) = x^{\frac{19}{7}}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{19}{7}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{19}{7}}]) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $n = - 1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{19}{7}}]) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.3.3

Replace all occurrences of $u_{2}$ with $x^{\frac{19}{7}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- {(x^{\frac{19}{7}})}^{- 2} \frac{d}{d x} [x^{\frac{19}{7}}]) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{19}{7}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- {(x^{\frac{19}{7}})}^{- 2} (\frac{19}{7} x^{\frac{19}{7} - 1})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.5

Multiply the exponents in ${(x^{\frac{19}{7}})}^{- 2}$ .

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Step 3.3.5.1

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

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{\frac{19}{7} \cdot - 2} (\frac{19}{7} x^{\frac{19}{7} - 1})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.5.2

Multiply $\frac{19}{7} \cdot - 2$ .

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Step 3.3.5.2.1

Combine $\frac{19}{7}$ and $- 2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{\frac{19 \cdot - 2}{7}} (\frac{19}{7} x^{\frac{19}{7} - 1})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.5.2.2

Multiply $19$ by $- 2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{\frac{- 38}{7}} (\frac{19}{7} x^{\frac{19}{7} - 1})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.5.3

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{19}{7} - 1})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{19}{7} - 1 \cdot \frac{7}{7}})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.7

Combine $- 1$ and $\frac{7}{7}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{19}{7} + \frac{- 1 \cdot 7}{7}})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.8

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{19 - 1 \cdot 7}{7}})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.9

Simplify the numerator.

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Step 3.3.9.1

Multiply $- 1$ by $7$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{19 - 7}{7}})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.9.2

Subtract $7$ from $19$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} (\frac{19}{7} x^{\frac{12}{7}})) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.10

Combine $\frac{19}{7}$ and $x^{\frac{12}{7}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- x^{- \frac{38}{7}} \frac{19 x^{\frac{12}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.11

Combine $\frac{19 x^{\frac{12}{7}}}{7}$ and $x^{- \frac{38}{7}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 x^{\frac{12}{7}} x^{- \frac{38}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.12

Multiply $x^{\frac{12}{7}}$ by $x^{- \frac{38}{7}}$ by adding the exponents.

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Step 3.3.12.1

Move $x^{- \frac{38}{7}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 (x^{- \frac{38}{7}} x^{\frac{12}{7}})}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.12.2

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

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 x^{- \frac{38}{7} + \frac{12}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.12.3

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 x^{\frac{- 38 + 12}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.12.4

Add $- 38$ and $12$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 x^{\frac{- 26}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.12.5

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19 x^{- \frac{26}{7}}}{7}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.13

Move $x^{- \frac{26}{7}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} + \frac{60}{7} (- \frac{19}{7 x^{\frac{26}{7}}}) + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.14

Multiply $\frac{60}{7}$ by $\frac{19}{7 x^{\frac{26}{7}}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{60 \cdot 19}{7 (7 x^{\frac{26}{7}})} + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.15

Multiply $60$ by $19$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{7 (7 x^{\frac{26}{7}})} + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.3.16

Multiply $7$ by $7$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4

Evaluate $\frac{d}{d x} [\frac{7}{9 x^{\frac{13}{6}}}]$ .

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Step 3.4.1

Since $\frac{7}{9}$ is constant with respect to $x$ , the derivative of $\frac{7}{9 x^{\frac{13}{6}}}$ with respect to $x$ is $\frac{7}{9} \frac{d}{d x} [\frac{1}{x^{\frac{13}{6}}}]$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} \frac{d}{d x} [\frac{1}{x^{\frac{13}{6}}}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.2

Rewrite $\frac{1}{x^{\frac{13}{6}}}$ as ${(x^{\frac{13}{6}})}^{- 1}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} \frac{d}{d x} [{(x^{\frac{13}{6}})}^{- 1}] + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $g (x) = x^{\frac{13}{6}}$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{13}{6}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

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 $n = - 1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.3.3

Replace all occurrences of $u_{3}$ with $x^{\frac{13}{6}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- {(x^{\frac{13}{6}})}^{- 2} \frac{d}{d x} [x^{\frac{13}{6}}]) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{13}{6}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- {(x^{\frac{13}{6}})}^{- 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5

Multiply the exponents in ${(x^{\frac{13}{6}})}^{- 2}$ .

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Step 3.4.5.1

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{13}{6} \cdot - 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.2

Cancel the common factor of $2$ .

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Step 3.4.5.2.1

Factor $2$ out of $6$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{13}{2 (3)} \cdot - 2} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.2.2

Factor $2$ out of $- 2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{13}{2 \cdot 3} \cdot (2 \cdot - 1)} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.2.3

Cancel the common factor.

Step 3.4.5.2.4

Rewrite the expression.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{13}{3} \cdot - 1} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.3

Combine $\frac{13}{3}$ and $- 1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{13 \cdot - 1}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.4

Multiply $13$ by $- 1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{\frac{- 13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.5.5

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} - 1 \cdot \frac{6}{6}})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.7

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13}{6} + \frac{- 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.8

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13 - 1 \cdot 6}{6}})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Multiply $- 1$ by $6$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{13 - 6}{6}})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.9.2

Subtract $6$ from $13$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} (\frac{13}{6} x^{\frac{7}{6}})) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.10

Combine $\frac{13}{6}$ and $x^{\frac{7}{6}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- x^{- \frac{13}{3}} \frac{13 x^{\frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.11

Combine $\frac{13 x^{\frac{7}{6}}}{6}$ and $x^{- \frac{13}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{\frac{7}{6}} x^{- \frac{13}{3}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12

Multiply $x^{\frac{7}{6}}$ by $x^{- \frac{13}{3}}$ by adding the exponents.

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Step 3.4.12.1

Move $x^{- \frac{13}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 (x^{- \frac{13}{3}} x^{\frac{7}{6}})}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.2

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{- \frac{13}{3} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.3

To write $- \frac{13}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{- \frac{13}{3} \cdot \frac{2}{2} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.12.4.1

Multiply $\frac{13}{3}$ by $\frac{2}{2}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{- \frac{13 \cdot 2}{3 \cdot 2} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.4.2

Multiply $3$ by $2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{- \frac{13 \cdot 2}{6} + \frac{7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.5

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{\frac{- 13 \cdot 2 + 7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.6

Simplify the numerator.

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Step 3.4.12.6.1

Multiply $- 13$ by $2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{\frac{- 26 + 7}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.6.2

Add $- 26$ and $7$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{\frac{- 19}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.12.7

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13 x^{- \frac{19}{6}}}{6}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.13

Move $x^{- \frac{19}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} + \frac{7}{9} (- \frac{13}{6 x^{\frac{19}{6}}}) + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.14

Multiply $\frac{7}{9}$ by $\frac{13}{6 x^{\frac{19}{6}}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{7 \cdot 13}{9 (6 x^{\frac{19}{6}})} + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.15

Multiply $7$ by $13$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{9 (6 x^{\frac{19}{6}})} + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.4.16

Multiply $6$ by $9$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + \frac{d}{d x} [- \frac{4}{3 x^{\frac{4}{3}}}]$

Step 3.5

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

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Step 3.5.1

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{4}{3}}}]$

Step 3.5.2

Rewrite $\frac{1}{x^{\frac{4}{3}}}$ as ${(x^{\frac{4}{3}})}^{- 1}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} \frac{d}{d x} [{(x^{\frac{4}{3}})}^{- 1}]$

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 $g (x) = x^{\frac{4}{3}}$ .

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Step 3.5.3.1

To apply the Chain Rule, set $u_{4}$ as $x^{\frac{4}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (\frac{d}{d u_{4}} [{u_{4}}^{- 1}] \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.5.3.2

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- {u_{4}}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.5.3.3

Replace all occurrences of $u_{4}$ with $x^{\frac{4}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- {(x^{\frac{4}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{4}{3}}])$

Step 3.5.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- {(x^{\frac{4}{3}})}^{- 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.5.5

Multiply the exponents in ${(x^{\frac{4}{3}})}^{- 2}$ .

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Step 3.5.5.1

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{\frac{4}{3} \cdot - 2} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.5.5.2

Multiply $\frac{4}{3} \cdot - 2$ .

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Step 3.5.5.2.1

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{\frac{4 \cdot - 2}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.5.5.2.2

Multiply $4$ by $- 2$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{\frac{- 8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.5.5.3

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1}))$

Step 3.5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}}))$

Step 3.5.7

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 3.5.8

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}}))$

Step 3.5.9

Simplify the numerator.

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Step 3.5.9.1

Multiply $- 1$ by $3$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{4 - 3}{3}}))$

Step 3.5.9.2

Subtract $3$ from $4$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} (\frac{4}{3} x^{\frac{1}{3}}))$

Step 3.5.10

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- x^{- \frac{8}{3}} \frac{4 x^{\frac{1}{3}}}{3})$

Step 3.5.11

Combine $\frac{4 x^{\frac{1}{3}}}{3}$ and $x^{- \frac{8}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 x^{\frac{1}{3}} x^{- \frac{8}{3}}}{3})$

Step 3.5.12

Multiply $x^{\frac{1}{3}}$ by $x^{- \frac{8}{3}}$ by adding the exponents.

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Step 3.5.12.1

Move $x^{- \frac{8}{3}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 (x^{- \frac{8}{3}} x^{\frac{1}{3}})}{3})$

Step 3.5.12.2

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

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 x^{- \frac{8}{3} + \frac{1}{3}}}{3})$

Step 3.5.12.3

Combine the numerators over the common denominator.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 x^{\frac{- 8 + 1}{3}}}{3})$

Step 3.5.12.4

Add $- 8$ and $1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 x^{\frac{- 7}{3}}}{3})$

Step 3.5.12.5

Move the negative in front of the fraction.

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4 x^{- \frac{7}{3}}}{3})$

Step 3.5.13

Move $x^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} - \frac{4}{3} (- \frac{4}{3 x^{\frac{7}{3}}})$

Step 3.5.14

Multiply $- 1$ by $- 1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + 1 (\frac{4}{3}) \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.5.15

Multiply $\frac{4}{3}$ by $1$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + \frac{4}{3} \cdot \frac{4}{3 x^{\frac{7}{3}}}$

Step 3.5.16

Multiply $\frac{4}{3}$ by $\frac{4}{3 x^{\frac{7}{3}}}$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + \frac{4 \cdot 4}{3 (3 x^{\frac{7}{3}})}$

Step 3.5.17

Multiply $4$ by $4$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + \frac{16}{3 (3 x^{\frac{7}{3}})}$

Step 3.5.18

Multiply $3$ by $3$ .

$- \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}} + \frac{16}{9 x^{\frac{7}{3}}}$

Step 3.6

Reorder terms.

$f''' (x) = \frac{16}{9 x^{\frac{7}{3}}} - \frac{96}{25 x^{\frac{11}{5}}} - \frac{1140}{49 x^{\frac{26}{7}}} - \frac{91}{54 x^{\frac{19}{6}}}$