Algebra Examples

$v = {(\sqrt{x} + \frac{1}{\sqrt[3]{x}})}^{2}$

Step 1

Rewrite the right side with rational exponents.

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Step 1.1

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

$x' = {(x^{\frac{1}{2}} + \frac{1}{\sqrt[3]{x}})}^{2}$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$x' = {(x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}})}^{2}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x'} x' = \frac{d}{d x'} {(x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}})}^{2}$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

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

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

Step 4.2

Expand $(x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}}) (x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}})$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

$\frac{d}{d v} [x^{\frac{1}{2}} x^{\frac{1}{2}} + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

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

$\frac{d}{d v} [x^{\frac{1}{2} + \frac{1}{2}} + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.1.2

Combine the numerators over the common denominator.

$\frac{d}{d v} [x^{\frac{1 + 1}{2}} + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.1.3

Add $1$ and $1$ .

$\frac{d}{d v} [x^{\frac{2}{2}} + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.1.4

Divide $2$ by $2$ .

$\frac{d}{d v} [x^{1} + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.2

Simplify $x^{1}$ .

$\frac{d}{d v} [x + x^{\frac{1}{2}} \frac{1}{x^{\frac{1}{3}}} + \frac{1}{x^{\frac{1}{3}}} x^{\frac{1}{2}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.3

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 4.3.1.3.1

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{2}}$ .

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

Step 4.3.1.3.2

Cancel the common factor.

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

Step 4.3.1.3.3

Rewrite the expression.

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

Step 4.3.1.4

Cancel the common factor of $x^{\frac{1}{3}}$ .

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Step 4.3.1.4.1

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{2}}$ .

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

Step 4.3.1.4.2

Cancel the common factor.

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

Step 4.3.1.4.3

Rewrite the expression.

$\frac{d}{d v} [x + x^{\frac{1}{6}} + x^{\frac{1}{6}} + \frac{1}{x^{\frac{1}{3}}} \cdot \frac{1}{x^{\frac{1}{3}}}]$

Step 4.3.1.5

Combine.

$\frac{d}{d v} [x + x^{\frac{1}{6}} + x^{\frac{1}{6}} + \frac{1 \cdot 1}{x^{\frac{1}{3}} x^{\frac{1}{3}}}]$

Step 4.3.1.6

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

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Step 4.3.1.6.1

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

$\frac{d}{d v} [x + x^{\frac{1}{6}} + x^{\frac{1}{6}} + \frac{1 \cdot 1}{x^{\frac{1}{3} + \frac{1}{3}}}]$

Step 4.3.1.6.2

Combine the numerators over the common denominator.

$\frac{d}{d v} [x + x^{\frac{1}{6}} + x^{\frac{1}{6}} + \frac{1 \cdot 1}{x^{\frac{1 + 1}{3}}}]$

Step 4.3.1.6.3

Add $1$ and $1$ .

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

Step 4.3.1.7

Multiply $1$ by $1$ .

$\frac{d}{d v} [x + x^{\frac{1}{6}} + x^{\frac{1}{6}} + \frac{1}{x^{\frac{2}{3}}}]$

Step 4.3.2

Add $x^{\frac{1}{6}}$ and $x^{\frac{1}{6}}$ .

$\frac{d}{d v} [x + 2 x^{\frac{1}{6}} + \frac{1}{x^{\frac{2}{3}}}]$

Step 4.4

By the Sum Rule, the derivative of $x + 2 x^{\frac{1}{6}} + \frac{1}{x^{\frac{2}{3}}}$ with respect to $x'$ is $\frac{d}{d v} [x] + \frac{d}{d v} [2 x^{\frac{1}{6}}] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$ .

$\frac{d}{d v} [x] + \frac{d}{d v} [2 x^{\frac{1}{6}}] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.5

Rewrite $\frac{d}{d v} [x]$ as $x'$ .

$x' + \frac{d}{d v} [2 x^{\frac{1}{6}}] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.6

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

$x' + 2 \frac{d}{d v} [x^{\frac{1}{6}}] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (x'))]$ is $f' (g (x')) g' (x')$ where $f (x') = {x'}^{\frac{1}{6}}$ and $g (x') = x$ .

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Step 4.7.1

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

$x' + 2 (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{6}}] \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.7.2

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

$x' + 2 (\frac{1}{6} {u_{1}}^{\frac{1}{6} - 1} \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.7.3

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

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

Step 4.8

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

$x' + 2 (\frac{1}{6} x^{\frac{1}{6} - 1 \cdot \frac{6}{6}} \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.9

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

$x' + 2 (\frac{1}{6} x^{\frac{1}{6} + \frac{- 1 \cdot 6}{6}} \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.10

Combine the numerators over the common denominator.

$x' + 2 (\frac{1}{6} x^{\frac{1 - 1 \cdot 6}{6}} \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.11

Simplify the numerator.

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Step 4.11.1

Multiply $- 1$ by $6$ .

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

Step 4.11.2

Subtract $6$ from $1$ .

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

Step 4.12

Move the negative in front of the fraction.

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

Step 4.13

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

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

Step 4.14

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

$x' + 2 (\frac{1}{6 x^{\frac{5}{6}}} \frac{d}{d v} [x]) + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.15

Combine $\frac{1}{6 x^{\frac{5}{6}}}$ and $2$ .

$x' + \frac{2}{6 x^{\frac{5}{6}}} \frac{d}{d v} [x] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.16

Factor $2$ out of $2$ .

$x' + \frac{2 (1)}{6 x^{\frac{5}{6}}} \frac{d}{d v} [x] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.17

Cancel the common factors.

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Step 4.17.1

Factor $2$ out of $6 x^{\frac{5}{6}}$ .

$x' + \frac{2 (1)}{2 (3 x^{\frac{5}{6}})} \frac{d}{d v} [x] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.17.2

Cancel the common factor.

$x' + \frac{2 \cdot 1}{2 (3 x^{\frac{5}{6}})} \frac{d}{d v} [x] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.17.3

Rewrite the expression.

$x' + \frac{1}{3 x^{\frac{5}{6}}} \frac{d}{d v} [x] + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.18

Rewrite $\frac{d}{d v} [x]$ as $x'$ .

$x' + \frac{1}{3 x^{\frac{5}{6}}} x' + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.19

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{d}{d v} [\frac{1}{x^{\frac{2}{3}}}]$

Step 4.20

Differentiate using the Quotient Rule which states that $\frac{d}{d v} [\frac{f (x')}{g (x')}]$ is $\frac{g (x') \frac{d}{d v} [f (x')] - f (x') \frac{d}{d v} [g (x')]}{{g (x')}^{2}}$ where $f (x') = 1$ and $g (x') = x^{\frac{2}{3}}$ .

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{x^{\frac{2}{3}} \frac{d}{d v} [1] - 1 \cdot 1 \frac{d}{d v} [x^{\frac{2}{3}}]}{{(x^{\frac{2}{3}})}^{2}}$

Step 4.21

Differentiate using the Constant Rule.

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Step 4.21.1

Multiply $- 1$ by $1$ .

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

Step 4.21.2

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

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Step 4.21.2.1

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{x^{\frac{2}{3}} \frac{d}{d v} [1] - \frac{d}{d v} [x^{\frac{2}{3}}]}{x^{\frac{2}{3} \cdot 2}}$

Step 4.21.2.2

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

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Step 4.21.2.2.1

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{x^{\frac{2}{3}} \frac{d}{d v} [1] - \frac{d}{d v} [x^{\frac{2}{3}}]}{x^{\frac{2 \cdot 2}{3}}}$

Step 4.21.2.2.2

Multiply $2$ by $2$ .

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

Step 4.21.3

Since $1$ is constant with respect to $x'$ , the derivative of $1$ with respect to $x'$ is $0$ .

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{x^{\frac{2}{3}} \cdot 0 - \frac{d}{d v} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 4.21.4

Simplify the expression.

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Step 4.21.4.1

Multiply $x^{\frac{2}{3}}$ by $0$ .

$x' + \frac{x'}{3 x^{\frac{5}{6}}} + \frac{0 - \frac{d}{d v} [x^{\frac{2}{3}}]}{x^{\frac{4}{3}}}$

Step 4.21.4.2

Subtract $\frac{d}{d v} [x^{\frac{2}{3}}]$ from $0$ .

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

Step 4.21.4.3

Move the negative in front of the fraction.

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

Step 4.22

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (x'))]$ is $f' (g (x')) g' (x')$ where $f (x') = {x'}^{\frac{2}{3}}$ and $g (x') = x$ .

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Step 4.22.1

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

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

Step 4.22.2

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

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

Step 4.22.3

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

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

Step 4.23

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

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

Step 4.24

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

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

Step 4.25

Combine the numerators over the common denominator.

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

Step 4.26

Simplify the numerator.

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Step 4.26.1

Multiply $- 1$ by $3$ .

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

Step 4.26.2

Subtract $3$ from $2$ .

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

Step 4.27

Move the negative in front of the fraction.

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

Step 4.28

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

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

Step 4.29

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

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

Step 4.30

Rewrite $\frac{d}{d v} [x]$ as $x'$ .

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{\frac{2}{3 x^{\frac{1}{3}}} x'}{x^{\frac{4}{3}}}$

Step 4.31

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{\frac{2 x'}{3 x^{\frac{1}{3}}}}{x^{\frac{4}{3}}}$

Step 4.32

Rewrite $\frac{\frac{2 x'}{3 x^{\frac{1}{3}}}}{x^{\frac{4}{3}}}$ as a product.

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

Step 4.33

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{1}{3}} x^{\frac{4}{3}}}$

Step 4.34

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{4}{3}}$ by adding the exponents.

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Step 4.34.1

Move $x^{\frac{4}{3}}$ .

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

Step 4.34.2

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

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{4}{3} + \frac{1}{3}}}$

Step 4.34.3

Combine the numerators over the common denominator.

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{4 + 1}{3}}}$

Step 4.34.4

Add $4$ and $1$ .

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

Step 5

Reform the equation by setting the left side equal to the right side.

$1 = x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{5}{3}}}$

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{5}{3}}} = 1$ .

$x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{5}{3}}} = 1$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 3 x^{\frac{5}{6}}, 3 x^{\frac{5}{3}}, 1$

Step 6.2.2

Since $1, 3 x^{\frac{5}{6}}, 3 x^{\frac{5}{3}}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 3, 3, 1$ then find LCM for the variable part $x^{\frac{5}{6}}, x^{\frac{5}{3}}$ .

Step 6.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.2.5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 6.2.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.2.7

The LCM of $1, 3, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 6.2.8

The LCM of $x^{\frac{5}{6}}, x^{\frac{5}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{5}{3}}$

Step 6.2.9

The LCM for $1, 3 x^{\frac{5}{6}}, 3 x^{\frac{5}{3}}, 1$ is the numeric part $3$ multiplied by the variable part.

$3 x^{\frac{5}{3}}$

Step 6.3

Multiply each term in $x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{5}{3}}} = 1$ by $3 x^{\frac{5}{3}}$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $x' + \frac{x'}{3 x^{\frac{5}{6}}} - \frac{2 x'}{3 x^{\frac{5}{3}}} = 1$ by $3 x^{\frac{5}{3}}$ .

$x' (3 x^{\frac{5}{3}}) + \frac{x'}{3 x^{\frac{5}{6}}} (3 x^{\frac{5}{3}}) - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Rewrite using the commutative property of multiplication.

$3 x' x^{\frac{5}{3}} + \frac{x'}{3 x^{\frac{5}{6}}} (3 x^{\frac{5}{3}}) - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.2

Rewrite using the commutative property of multiplication.

$3 x' x^{\frac{5}{3}} + 3 \frac{x'}{3 x^{\frac{5}{6}}} x^{\frac{5}{3}} - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.3

Cancel the common factor of $3$ .

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Step 6.3.2.1.3.1

Cancel the common factor.

$3 x' x^{\frac{5}{3}} + 3 \frac{x'}{3 x^{\frac{5}{6}}} x^{\frac{5}{3}} - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.3.2

Rewrite the expression.

$3 x' x^{\frac{5}{3}} + \frac{x'}{x^{\frac{5}{6}}} x^{\frac{5}{3}} - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.4

Cancel the common factor of $x^{\frac{5}{6}}$ .

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Step 6.3.2.1.4.1

Factor $x^{\frac{5}{6}}$ out of $x^{\frac{5}{3}}$ .

$3 x' x^{\frac{5}{3}} + \frac{x'}{x^{\frac{5}{6}}} (x^{\frac{5}{6}} x^{\frac{5}{6}}) - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.4.2

Cancel the common factor.

$3 x' x^{\frac{5}{3}} + \frac{x'}{x^{\frac{5}{6}}} (x^{\frac{5}{6}} x^{\frac{5}{6}}) - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.4.3

Rewrite the expression.

$3 x' x^{\frac{5}{3}} + x' x^{\frac{5}{6}} - \frac{2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.5

Cancel the common factor of $3 x^{\frac{5}{3}}$ .

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Step 6.3.2.1.5.1

Move the leading negative in $- \frac{2 x'}{3 x^{\frac{5}{3}}}$ into the numerator.

$3 x' x^{\frac{5}{3}} + x' x^{\frac{5}{6}} + \frac{- 2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.5.2

Cancel the common factor.

$3 x' x^{\frac{5}{3}} + x' x^{\frac{5}{6}} + \frac{- 2 x'}{3 x^{\frac{5}{3}}} (3 x^{\frac{5}{3}}) = 1 (3 x^{\frac{5}{3}})$

Step 6.3.2.1.5.3

Rewrite the expression.

$3 x' x^{\frac{5}{3}} + x' x^{\frac{5}{6}} - 2 x' = 1 (3 x^{\frac{5}{3}})$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

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

Step 6.4

Solve the equation.

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Step 6.4.1

Factor $x'$ out of $3 x' x^{\frac{5}{3}} + x' x^{\frac{5}{6}} - 2 x'$ .

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Step 6.4.1.1

Factor $x'$ out of $3 x' x^{\frac{5}{3}}$ .

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

Step 6.4.1.2

Factor $x'$ out of $x' x^{\frac{5}{6}}$ .

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

Step 6.4.1.3

Factor $x'$ out of $- 2 x'$ .

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

Step 6.4.1.4

Factor $x'$ out of $x' (3 x^{\frac{5}{3}}) + x' (x^{\frac{5}{6}})$ .

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

Step 6.4.1.5

Factor $x'$ out of $x' (3 x^{\frac{5}{3}} + x^{\frac{5}{6}}) + x' \cdot - 2$ .

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

Step 6.4.2

Rewrite $x^{\frac{5}{3}}$ as ${(x^{\frac{5}{6}})}^{2}$ .

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

Step 6.4.3

Let $u = x^{\frac{5}{6}}$ . Substitute $u$ for all occurrences of $x^{\frac{5}{6}}$ .

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

Step 6.4.4

Factor by grouping.

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Step 6.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Step 6.4.4.1.1

Multiply by $1$ .

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

Step 6.4.4.1.2

Rewrite $1$ as $- 2$ plus $3$

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

Step 6.4.4.1.3

Apply the distributive property.

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

Step 6.4.4.2

Factor out the greatest common factor from each group.

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Step 6.4.4.2.1

Group the first two terms and the last two terms.

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

Step 6.4.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 6.4.4.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 2$ .

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

Step 6.4.5

Factor.

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Step 6.4.5.1

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

$x' ((3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)) = 3 x^{\frac{5}{3}}$

Step 6.4.5.2

Remove unnecessary parentheses.

$x' (3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1) = 3 x^{\frac{5}{3}}$

Step 6.4.6

Divide each term in $x' (3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1) = 3 x^{\frac{5}{3}}$ by $(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)$ and simplify.

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Step 6.4.6.1

Divide each term in $x' (3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1) = 3 x^{\frac{5}{3}}$ by $(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)$ .

$\frac{x' (3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)} = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$

Step 6.4.6.2

Simplify the left side.

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Step 6.4.6.2.1

Cancel the common factor.

$\frac{x' (3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)} = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$

Step 6.4.6.2.2

Rewrite the expression.

$\frac{x' (x^{\frac{5}{6}} + 1)}{x^{\frac{5}{6}} + 1} = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$

Step 6.4.6.2.3

Cancel the common factor.

$\frac{x' (x^{\frac{5}{6}} + 1)}{x^{\frac{5}{6}} + 1} = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$

Step 6.4.6.2.4

Divide $x'$ by $1$ .

$x' = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$

Step 7

Replace $x'$ with $\frac{d x}{d v}$ .

$\frac{d x}{d v} = \frac{3 x^{\frac{5}{3}}}{(3 x^{\frac{5}{6}} - 2) (x^{\frac{5}{6}} + 1)}$