Precalculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $3$ by $2$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{1}{3}}$ and $g (y) = 1 + 2 x$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $3$ .

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

Step 3.7.2

Subtract $3$ from $1$ .

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

Step 3.8

Combine fractions.

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Step 3.8.1

Move the negative in front of the fraction.

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

Step 3.8.2

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

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

Step 3.8.3

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

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

Step 3.8.4

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

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

Step 3.9

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

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

Step 3.10

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

$\frac{\frac{{(1 - x)}^{3}}{3 {(1 + 2 x)}^{\frac{2}{3}}} (0 + \frac{d}{d y} [2 x]) - {(1 + 2 x)}^{\frac{1}{3}} \frac{d}{d y} [{(1 - x)}^{3}]}{{(1 - x)}^{6}}$

Step 3.11

Add $0$ and $\frac{d}{d y} [2 x]$ .

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Combine $\frac{2 {(1 - x)}^{3}}{3 {(1 + 2 x)}^{\frac{2}{3}}}$ and $x'$ .

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

Step 3.16

Multiply $\frac{\frac{2 {(1 - x)}^{3} x'}{3 {(1 + 2 x)}^{\frac{2}{3}}} - {(1 + 2 x)}^{\frac{1}{3}} \frac{d}{d y} [{(1 - x)}^{3}]}{{(1 - x)}^{6}}$ by $\frac{3 {(1 + 2 x)}^{\frac{2}{3}}}{3 {(1 + 2 x)}^{\frac{2}{3}}}$ .

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

Step 3.17

Combine.

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

Step 3.18

Apply the distributive property.

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

Step 3.19

Cancel the common factor of $3 {(1 + 2 x)}^{\frac{2}{3}}$ .

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Step 3.19.1

Cancel the common factor.

Step 3.19.2

Rewrite the expression.

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

Step 3.20

Multiply $- 1$ by $3$ .

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

Step 3.21

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

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

Step 3.22

Simplify the expression.

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Step 3.22.1

Combine the numerators over the common denominator.

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

Step 3.22.2

Add $1$ and $2$ .

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

Step 3.23

Cancel the common factor of $3$ .

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Step 3.23.1

Cancel the common factor.

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

Step 3.23.2

Rewrite the expression.

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

Step 3.24

Simplify.

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

Step 3.25

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

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Step 3.25.1

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

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

Step 3.25.2

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

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

Step 3.25.3

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

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

Step 3.26

Differentiate.

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Step 3.26.1

Multiply $3$ by $- 3$ .

$\frac{2 {(1 - x)}^{3} x' - 9 (1 + 2 x) ({(1 - x)}^{2} \frac{d}{d y} [1 - x])}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.26.2

By the Sum Rule, the derivative of $1 - x$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- x]$ .

$\frac{2 {(1 - x)}^{3} x' - 9 (1 + 2 x) ({(1 - x)}^{2} (\frac{d}{d y} [1] + \frac{d}{d y} [- x]))}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.26.3

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

$\frac{2 {(1 - x)}^{3} x' - 9 (1 + 2 x) ({(1 - x)}^{2} (0 + \frac{d}{d y} [- x]))}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.26.4

Add $0$ and $\frac{d}{d y} [- x]$ .

$\frac{2 {(1 - x)}^{3} x' - 9 (1 + 2 x) ({(1 - x)}^{2} \frac{d}{d y} [- x])}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.26.5

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

$\frac{2 {(1 - x)}^{3} x' - 9 (1 + 2 x) ({(1 - x)}^{2} (- \frac{d}{d y} [x]))}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.26.6

Multiply $- 1$ by $- 9$ .

$\frac{2 {(1 - x)}^{3} x' + 9 (1 + 2 x) ({(1 - x)}^{2} (\frac{d}{d y} [x]))}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.27

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

$\frac{2 {(1 - x)}^{3} x' + 9 (1 + 2 x) ({(1 - x)}^{2} x')}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28

Simplify.

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Step 3.28.1

Apply the distributive property.

$\frac{2 {(1 - x)}^{3} x' + (9 \cdot 1 + 9 (2 x)) ({(1 - x)}^{2} x')}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2

Simplify the numerator.

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Step 3.28.2.1

Factor ${(1 - x)}^{2} x'$ out of $2 {(1 - x)}^{3} x' + (9 \cdot 1 + 9 \cdot 2 x) {(1 - x)}^{2} x'$ .

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Step 3.28.2.1.1

Factor ${(1 - x)}^{2} x'$ out of $2 {(1 - x)}^{3} x'$ .

$\frac{{(1 - x)}^{2} x' (2 (1 - x)) + (9 \cdot 1 + 9 \cdot 2 x) {(1 - x)}^{2} x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.1.2

Factor ${(1 - x)}^{2} x'$ out of $(9 \cdot 1 + 9 \cdot 2 x) {(1 - x)}^{2} x'$ .

$\frac{{(1 - x)}^{2} x' (2 (1 - x)) + {(1 - x)}^{2} x' (9 \cdot 1 + 9 \cdot 2 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.1.3

Factor ${(1 - x)}^{2} x'$ out of ${(1 - x)}^{2} x' (2 (1 - x)) + {(1 - x)}^{2} x' (9 \cdot 1 + 9 \cdot 2 x)$ .

$\frac{{(1 - x)}^{2} x' (2 (1 - x) + 9 \cdot 1 + 9 \cdot 2 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.2

Combine exponents.

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Step 3.28.2.2.1

Multiply $9$ by $1$ .

$\frac{{(1 - x)}^{2} x' (2 (1 - x) + 9 + 9 \cdot 2 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.2.2

Multiply $9$ by $2$ .

$\frac{{(1 - x)}^{2} x' (2 (1 - x) + 9 + 18 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.3

Simplify each term.

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Step 3.28.2.3.1

Apply the distributive property.

$\frac{{(1 - x)}^{2} x' (2 \cdot 1 + 2 (- x) + 9 + 18 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.3.2

Multiply $2$ by $1$ .

$\frac{{(1 - x)}^{2} x' (2 + 2 (- x) + 9 + 18 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.3.3

Multiply $- 1$ by $2$ .

$\frac{{(1 - x)}^{2} x' (2 - 2 x + 9 + 18 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.4

Add $2$ and $9$ .

$\frac{{(1 - x)}^{2} x' (- 2 x + 11 + 18 x)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.2.5

Add $- 2 x$ and $18 x$ .

$\frac{{(1 - x)}^{2} x' (16 x + 11)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.3

Combine terms.

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Step 3.28.3.1

Factor ${(1 - x)}^{2}$ out of ${(1 - x)}^{2} x' (16 x + 11)$ .

$\frac{{(1 - x)}^{2} (x' (16 x + 11))}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}}$

Step 3.28.3.2

Cancel the common factors.

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Step 3.28.3.2.1

Factor ${(1 - x)}^{2}$ out of $3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{6}$ .

$\frac{{(1 - x)}^{2} (x' (16 x + 11))}{{(1 - x)}^{2} (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})}$

Step 3.28.3.2.2

Cancel the common factor.

$\frac{{(1 - x)}^{2} (x' (16 x + 11))}{{(1 - x)}^{2} (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})}$

Step 3.28.3.2.3

Rewrite the expression.

$\frac{x' (16 x + 11)}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}}$

Step 3.28.4

Reorder terms.

$\frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}}$

Step 4

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

$1 = \frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} = 1$ .

$\frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} = 1$

Step 5.2

Multiply both sides by $3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$ .

$\frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$ .

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Step 5.3.1.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{(16 x + 11) x'}{3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.2

Cancel the common factor of $3$ .

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Step 5.3.1.1.2.1

Factor $3$ out of $3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$ .

$3 \frac{(16 x + 11) x'}{3 ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})} ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.2.2

Cancel the common factor.

$3 \frac{(16 x + 11) x'}{3 ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})} ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.2.3

Rewrite the expression.

$\frac{(16 x + 11) x'}{{(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.3

Cancel the common factor of ${(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$ .

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Step 5.3.1.1.3.1

Cancel the common factor.

$\frac{(16 x + 11) x'}{{(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}} ({(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}) = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.3.2

Rewrite the expression.

$(16 x + 11) x' = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.4

Apply the distributive property.

$16 x x' + 11 x' = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.1.1.5

Move $x$ .

$16 x' x + 11 x' = 1 (3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$

Step 5.4

Solve for $x'$ .

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Step 5.4.1

Simplify $3 {(1 + 2 x)}^{\frac{2}{3}} {(1 - x)}^{4}$ .

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Step 5.4.1.1

Use the Binomial Theorem.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1^{4} + 4 \cdot 1^{3} (- x) + 6 \cdot 1^{2} {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2

Simplify terms.

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Step 5.4.1.2.1

Simplify each term.

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Step 5.4.1.2.1.1

One to any power is one.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 + 4 \cdot 1^{3} (- x) + 6 \cdot 1^{2} {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.2

One to any power is one.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 + 4 \cdot 1 (- x) + 6 \cdot 1^{2} {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.3

Multiply $4$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 + 4 (- x) + 6 \cdot 1^{2} {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.4

Multiply $- 1$ by $4$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 \cdot 1^{2} {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.5

One to any power is one.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 \cdot 1 {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.6

Multiply $6$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 {(- x)}^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.7

Apply the product rule to $- x$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 ({(- 1)}^{2} x^{2}) + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.8

Raise $- 1$ to the power of $2$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 (1 x^{2}) + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.9

Multiply $x^{2}$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} + 4 \cdot 1 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.10

Multiply $4$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} + 4 {(- x)}^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.11

Apply the product rule to $- x$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} + 4 ({(- 1)}^{3} x^{3}) + {(- x)}^{4})$

Step 5.4.1.2.1.12

Raise $- 1$ to the power of $3$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} + 4 (- x^{3}) + {(- x)}^{4})$

Step 5.4.1.2.1.13

Multiply $- 1$ by $4$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} - 4 x^{3} + {(- x)}^{4})$

Step 5.4.1.2.1.14

Apply the product rule to $- x$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} - 4 x^{3} + {(- 1)}^{4} x^{4})$

Step 5.4.1.2.1.15

Raise $- 1$ to the power of $4$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} - 4 x^{3} + 1 x^{4})$

Step 5.4.1.2.1.16

Multiply $x^{4}$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} (1 - 4 x + 6 x^{2} - 4 x^{3} + x^{4})$

Step 5.4.1.2.2

Apply the distributive property.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} \cdot 1 + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x) + 3 {(1 + 2 x)}^{\frac{2}{3}} (6 x^{2}) + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x^{3}) + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.3

Simplify.

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Step 5.4.1.3.1

Multiply $3$ by $1$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x) + 3 {(1 + 2 x)}^{\frac{2}{3}} (6 x^{2}) + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x^{3}) + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.3.2

Rewrite using the commutative property of multiplication.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x + 3 {(1 + 2 x)}^{\frac{2}{3}} (6 x^{2}) + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x^{3}) + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.3.3

Rewrite using the commutative property of multiplication.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x + 3 \cdot 6 {(1 + 2 x)}^{\frac{2}{3}} x^{2} + 3 {(1 + 2 x)}^{\frac{2}{3}} (- 4 x^{3}) + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.3.4

Rewrite using the commutative property of multiplication.

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x + 3 \cdot 6 {(1 + 2 x)}^{\frac{2}{3}} x^{2} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x^{3} + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.4

Simplify each term.

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Step 5.4.1.4.1

Multiply $3$ by $- 4$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 {(1 + 2 x)}^{\frac{2}{3}} x + 3 \cdot 6 {(1 + 2 x)}^{\frac{2}{3}} x^{2} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x^{3} + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.4.2

Multiply $3$ by $6$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 {(1 + 2 x)}^{\frac{2}{3}} x + 18 {(1 + 2 x)}^{\frac{2}{3}} x^{2} + 3 \cdot - 4 {(1 + 2 x)}^{\frac{2}{3}} x^{3} + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.4.3

Multiply $3$ by $- 4$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 {(1 + 2 x)}^{\frac{2}{3}} x + 18 {(1 + 2 x)}^{\frac{2}{3}} x^{2} - 12 {(1 + 2 x)}^{\frac{2}{3}} x^{3} + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$

Step 5.4.1.5

Reorder factors in $3 {(1 + 2 x)}^{\frac{2}{3}} - 12 {(1 + 2 x)}^{\frac{2}{3}} x + 18 {(1 + 2 x)}^{\frac{2}{3}} x^{2} - 12 {(1 + 2 x)}^{\frac{2}{3}} x^{3} + 3 {(1 + 2 x)}^{\frac{2}{3}} x^{4}$ .

$16 x' x + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$

Step 5.4.2

Factor $x'$ out of $16 x' x + 11 x'$ .

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Step 5.4.2.1

Factor $x'$ out of $16 x' x$ .

$x' (16 x) + 11 x' = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$

Step 5.4.2.2

Factor $x'$ out of $11 x'$ .

$x' (16 x) + x' \cdot 11 = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$

Step 5.4.2.3

Factor $x'$ out of $x' (16 x) + x' \cdot 11$ .

$x' (16 x + 11) = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$

Step 5.4.3

Divide each term in $x' (16 x + 11) = 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$ by $16 x + 11$ and simplify.

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Step 5.4.3.1

$\frac{x' (16 x + 11)}{16 x + 11} = \frac{3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{- 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $16 x + 11$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

Step 5.4.3.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{- 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Simplify each term.

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Step 5.4.3.3.1.1

Move the negative in front of the fraction.

$x' = \frac{3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{- 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.1.2

Move the negative in front of the fraction.

$x' = \frac{3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2

Simplify terms.

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Step 5.4.3.3.2.1

Combine the numerators over the common denominator.

$x' = \frac{3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.2

Combine the numerators over the common denominator.

$x' = \frac{3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.3

Reorder terms.

$x' = \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} - \frac{12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.4

Combine the numerators over the common denominator.

$x' = \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11} + \frac{3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.5

Combine the numerators over the common denominator.

$x' = \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.6

Reorder terms.

$x' = \frac{- 12 x {(1 + 2 x)}^{\frac{2}{3}} + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.7

Factor $- 1$ out of $- 12 x {(1 + 2 x)}^{\frac{2}{3}}$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}}) + 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.8

Factor $- 1$ out of $18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}}) - (- 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}) - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.9

Factor $- 1$ out of $- (12 x {(1 + 2 x)}^{\frac{2}{3}}) - (- 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}})$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}) - 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.10

Factor $- 1$ out of $- 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}) - (12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}) + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.11

Factor $- 1$ out of $- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}}) - (12 x^{3} {(1 + 2 x)}^{\frac{2}{3}})$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}) + 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.12

Factor $- 1$ out of $3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}) - (- 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}) + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.13

Factor $- 1$ out of $- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}}) - (- 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}})$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}) + 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 5.4.3.3.2.14

Factor $- 1$ out of $3 {(1 + 2 x)}^{\frac{2}{3}}$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}) - (- 3 {(1 + 2 x)}^{\frac{2}{3}})}{16 x + 11}$

Step 5.4.3.3.2.15

Factor $- 1$ out of $- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}}) - (- 3 {(1 + 2 x)}^{\frac{2}{3}})$ .

$x' = \frac{- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}})}{16 x + 11}$

Step 5.4.3.3.2.16

Simplify the expression.

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Step 5.4.3.3.2.16.1

Rewrite $- (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}})$ as $- 1 (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}})$ .

$x' = \frac{- 1 (12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}})}{16 x + 11}$

Step 5.4.3.3.2.16.2

Move the negative in front of the fraction.

$x' = - \frac{12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$

Step 6

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

$\frac{d x}{d y} = - \frac{12 x {(1 + 2 x)}^{\frac{2}{3}} - 18 x^{2} {(1 + 2 x)}^{\frac{2}{3}} + 12 x^{3} {(1 + 2 x)}^{\frac{2}{3}} - 3 x^{4} {(1 + 2 x)}^{\frac{2}{3}} - 3 {(1 + 2 x)}^{\frac{2}{3}}}{16 x + 11}$