Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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

Step 3.2

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

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

Combine the numerators over the common denominator.

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

Step 3.3.3

Simplify the numerator.

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Step 3.3.3.1

Rewrite $1$ as $1^{3}$ .

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

Step 3.3.3.2

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

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Step 3.3.3.2.1

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

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Step 3.3.3.2.1.1

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

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

Step 3.3.3.2.1.2

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

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

Step 3.3.3.2.2

Add $1$ and $2$ .

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

Step 3.3.3.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = x$ .

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

Step 3.3.3.4

Simplify.

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Step 3.3.3.4.1

One to any power is one.

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

Step 3.3.3.4.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3.3.4

Apply the product rule to $\frac{(1 + x) (1 - x + x^{2})}{x^{2}}$ .

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

Step 3.3.5

Apply the product rule to $(1 + x) (1 - x + x^{2})$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

Multiply $2$ by $3$ .

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $\frac{x^{6}}{{(1 + x)}^{3} {(1 - x + x^{2})}^{3}}$ by $1$ .

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

Step 3.6

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 $f (y) = x^{6}$ and $g (y) = {(1 + x)}^{3} {(1 - x + x^{2})}^{3}$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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 $g (y) = 1 - x + x^{2}$ .

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Step 3.10.1

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

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

Step 3.10.2

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

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

Step 3.10.3

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

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

Step 3.11

Differentiate.

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Step 3.11.1

Move $3$ to the left of ${(1 + x)}^{3}$ .

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

Step 3.11.2

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

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

Step 3.11.3

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

$\frac{{(1 + x)}^{3} {(1 - x + x^{2})}^{3} (6 x^{5} x') - x^{6} (3 {(1 + x)}^{3} {(1 - x + x^{2})}^{2} (0 + \frac{d}{d y} [- x] + \frac{d}{d y} [x^{2}]) + {(1 - x + x^{2})}^{3} \frac{d}{d y} [{(1 + x)}^{3}])}{{({(1 + x)}^{3} {(1 - x + x^{2})}^{3})}^{2}}$

Step 3.11.4

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

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

Step 3.11.5

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

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

Step 3.12

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

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

Step 3.13

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

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Step 3.13.1

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

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

Step 3.13.2

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

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

Step 3.13.3

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

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

Step 3.14

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

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

Step 3.15

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 3.15.1

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

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

Step 3.15.2

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

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

Step 3.15.3

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

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

Step 3.16

Differentiate.

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Step 3.16.1

Move $3$ to the left of ${(1 - x + x^{2})}^{3}$ .

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

Step 3.16.2

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

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

Step 3.16.3

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

$\frac{{(1 + x)}^{3} {(1 - x + x^{2})}^{3} (6 x^{5} x') - x^{6} (3 {(1 + x)}^{3} {(1 - x + x^{2})}^{2} (- x' + 2 x x') + 3 {(1 - x + x^{2})}^{3} {(1 + x)}^{2} (0 + \frac{d}{d y} [x]))}{{({(1 + x)}^{3} {(1 - x + x^{2})}^{3})}^{2}}$

Step 3.16.4

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

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

Step 3.17

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

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

Step 3.18

Simplify.

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Step 3.18.1

Apply the product rule to ${(1 + x)}^{3} {(1 - x + x^{2})}^{3}$ .

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

Step 3.18.2

Apply the distributive property.

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

Step 3.18.3

Simplify the numerator.

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Step 3.18.3.1

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of ${(1 + x)}^{3} {(1 - x + x^{2})}^{3} (6 x^{5} x') - x^{6} (3 {(1 + x)}^{3} {(1 - x + x^{2})}^{2} (- x' + 2 x x')) - x^{6} (3 {(1 - x + x^{2})}^{3} {(1 + x)}^{2} x')$ .

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Step 3.18.3.1.1

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of ${(1 + x)}^{3} {(1 - x + x^{2})}^{3} (6 x^{5} x')$ .

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

Step 3.18.3.1.2

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of $- x^{6} (3 {(1 + x)}^{3} {(1 - x + x^{2})}^{2} (- x' + 2 x x'))$ .

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

Step 3.18.3.1.3

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of $- x^{6} (3 {(1 - x + x^{2})}^{3} {(1 + x)}^{2} x')$ .

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

Step 3.18.3.1.4

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (((1 + x) (1 - x + x^{2})) (6 x')) + {(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (- x (3 (1 + x) (- x' + 2 x x')))$ .

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

Step 3.18.3.1.5

Factor ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ out of ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (((1 + x) (1 - x + x^{2})) (6 x') - x (3 (1 + x) (- x' + 2 x x'))) + {(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (- x (3 (1 - x + x^{2}) x'))$ .

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

Step 3.18.3.2

Combine exponents.

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Step 3.18.3.2.1

Multiply $3$ by $- 1$ .

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

Step 3.18.3.2.2

Multiply $3$ by $- 1$ .

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

Step 3.18.3.3

Simplify each term.

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Step 3.18.3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.18.3.3.2

Apply the distributive property.

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

Step 3.18.3.3.3

Multiply $6$ by $1$ .

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

Step 3.18.3.3.4

Expand $(6 + 6 x) (1 - x + x^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.18.3.3.5

Combine the opposite terms in $6 \cdot 1 + 6 (- x) + 6 x^{2} + 6 x \cdot 1 + 6 x (- x) + 6 x \cdot x^{2}$ .

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Step 3.18.3.3.5.1

Reorder the factors in the terms $6 (- x)$ and $6 x \cdot 1$ .

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

Step 3.18.3.3.5.2

Add $- 1 \cdot 6 x$ and $1 \cdot 6 x$ .

$\frac{{(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} ((6 \cdot 1 + 6 x^{2} + 0 + 6 x (- x) + 6 x \cdot x^{2}) x' - 3 x ((1 + x) (- x' + 2 x x')) - 3 x ((1 - x + x^{2}) x'))}{{({(1 + x)}^{3})}^{2} {({(1 - x + x^{2})}^{3})}^{2}}$

Step 3.18.3.3.5.3

Add $6 \cdot 1 + 6 x^{2}$ and $0$ .

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

Step 3.18.3.3.6

Simplify each term.

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Step 3.18.3.3.6.1

Multiply $6$ by $1$ .

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

Step 3.18.3.3.6.2

Rewrite using the commutative property of multiplication.

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

Step 3.18.3.3.6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.18.3.3.6.3.1

Move $x$ .

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

Step 3.18.3.3.6.3.2

Multiply $x$ by $x$ .

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

Step 3.18.3.3.6.4

Multiply $6$ by $- 1$ .

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

Step 3.18.3.3.6.5

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

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Step 3.18.3.3.6.5.1

Move $x^{2}$ .

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

Step 3.18.3.3.6.5.2

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

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Step 3.18.3.3.6.5.2.1

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

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

Step 3.18.3.3.6.5.2.2

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

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

Step 3.18.3.3.6.5.3

Add $2$ and $1$ .

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

Step 3.18.3.3.7

Combine the opposite terms in $6 + 6 x^{2} - 6 x^{2} + 6 x^{3}$ .

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Step 3.18.3.3.7.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$\frac{{(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} ((6 + 0 + 6 x^{3}) x' - 3 x ((1 + x) (- x' + 2 x x')) - 3 x ((1 - x + x^{2}) x'))}{{({(1 + x)}^{3})}^{2} {({(1 - x + x^{2})}^{3})}^{2}}$

Step 3.18.3.3.7.2

Add $6$ and $0$ .

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

Step 3.18.3.3.8

Apply the distributive property.

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

Step 3.18.3.3.9

Expand $(1 + x) (- x' + 2 x x')$ using the FOIL Method.

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Step 3.18.3.3.9.1

Apply the distributive property.

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

Step 3.18.3.3.9.2

Apply the distributive property.

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

Step 3.18.3.3.9.3

Apply the distributive property.

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

Step 3.18.3.3.10

Simplify and combine like terms.

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Step 3.18.3.3.10.1

Simplify each term.

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Step 3.18.3.3.10.1.1

Multiply $- x'$ by $1$ .

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

Step 3.18.3.3.10.1.2

Multiply $2 x x'$ by $1$ .

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

Step 3.18.3.3.10.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.18.3.3.10.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.18.3.3.10.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.18.3.3.10.1.5.1

Move $x$ .

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

Step 3.18.3.3.10.1.5.2

Multiply $x$ by $x$ .

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

Step 3.18.3.3.10.2

Subtract $x x'$ from $2 x x'$ .

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

Step 3.18.3.3.11

Multiply $x$ by $1$ .

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

Step 3.18.3.3.12

Apply the distributive property.

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

Step 3.18.3.3.13

Simplify.

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Step 3.18.3.3.13.1

Multiply $- 1$ by $- 3$ .

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

Step 3.18.3.3.13.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.18.3.3.13.2.1

Move $x$ .

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

Step 3.18.3.3.13.2.2

Multiply $x$ by $x$ .

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

Step 3.18.3.3.13.3

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

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Step 3.18.3.3.13.3.1

Move $x^{2}$ .

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

Step 3.18.3.3.13.3.2

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

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Step 3.18.3.3.13.3.2.1

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

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

Step 3.18.3.3.13.3.2.2

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

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

Step 3.18.3.3.13.3.3

Add $2$ and $1$ .

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

Step 3.18.3.3.14

Multiply $2$ by $- 3$ .

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

Step 3.18.3.3.15

Apply the distributive property.

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

Step 3.18.3.3.16

Multiply $x'$ by $1$ .

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

Step 3.18.3.3.17

Apply the distributive property.

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

Step 3.18.3.3.18

Simplify.

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Step 3.18.3.3.18.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.18.3.3.18.1.1

Move $x$ .

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

Step 3.18.3.3.18.1.2

Multiply $x$ by $x$ .

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

Step 3.18.3.3.18.2

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

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Step 3.18.3.3.18.2.1

Move $x^{2}$ .

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

Step 3.18.3.3.18.2.2

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

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Step 3.18.3.3.18.2.2.1

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

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

Step 3.18.3.3.18.2.2.2

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

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

Step 3.18.3.3.18.2.3

Add $2$ and $1$ .

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

Step 3.18.3.3.19

Simplify each term.

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Step 3.18.3.3.19.1

Rewrite $- 1 x'$ as $- x'$ .

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

Step 3.18.3.3.19.2

Multiply $- 1$ by $- 3$ .

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

Step 3.18.3.4

Combine the opposite terms in $6 x' + 6 x^{3} x' + 3 x x' - 3 x^{2} x' - 6 x^{3} x' - 3 x x' + 3 x^{2} x' - 3 x^{3} x'$ .

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Step 3.18.3.4.1

Subtract $6 x^{3} x'$ from $6 x^{3} x'$ .

$\frac{{(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (6 x' + 3 x x' - 3 x^{2} x' + 0 - 3 x x' + 3 x^{2} x' - 3 x^{3} x')}{{({(1 + x)}^{3})}^{2} {({(1 - x + x^{2})}^{3})}^{2}}$

Step 3.18.3.4.2

Add $6 x' + 3 x x' - 3 x^{2} x'$ and $0$ .

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

Step 3.18.3.4.3

Subtract $3 x x'$ from $3 x x'$ .

$\frac{{(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (6 x' - 3 x^{2} x' + 0 + 3 x^{2} x' - 3 x^{3} x')}{{({(1 + x)}^{3})}^{2} {({(1 - x + x^{2})}^{3})}^{2}}$

Step 3.18.3.4.4

Add $6 x' - 3 x^{2} x'$ and $0$ .

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

Step 3.18.3.4.5

Add $- 3 x^{2} x'$ and $3 x^{2} x'$ .

$\frac{{(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} (6 x' + 0 - 3 x^{3} x')}{{({(1 + x)}^{3})}^{2} {({(1 - x + x^{2})}^{3})}^{2}}$

Step 3.18.3.4.6

Add $6 x'$ and $0$ .

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

Step 3.18.3.5

Factor $3 x'$ out of $6 x' - 3 x^{3} x'$ .

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Step 3.18.3.5.1

Factor $3 x'$ out of $6 x'$ .

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

Step 3.18.3.5.2

Factor $3 x'$ out of $- 3 x^{3} x'$ .

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

Step 3.18.3.5.3

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

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

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

Step 3.18.4

Combine terms.

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Step 3.18.4.1

Move $3$ to the left of ${(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5}$ .

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

Step 3.18.4.2

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

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Step 3.18.4.2.1

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

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

Step 3.18.4.2.2

Multiply $3$ by $2$ .

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

Step 3.18.4.3

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

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Step 3.18.4.3.1

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

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

Step 3.18.4.3.2

Multiply $3$ by $2$ .

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

Step 3.18.4.4

Cancel the common factor of ${(1 + x)}^{2}$ and ${(1 + x)}^{6}$ .

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Step 3.18.4.4.1

Factor ${(1 + x)}^{2}$ out of $3 {(1 + x)}^{2} {(1 - x + x^{2})}^{2} x^{5} x' (2 - x^{3})$ .

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

Step 3.18.4.4.2

Cancel the common factors.

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Step 3.18.4.4.2.1

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

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

Step 3.18.4.4.2.2

Cancel the common factor.

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

Step 3.18.4.4.2.3

Rewrite the expression.

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

Step 3.18.4.5

Cancel the common factor of ${(1 - x + x^{2})}^{2}$ and ${(1 - x + x^{2})}^{6}$ .

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Step 3.18.4.5.1

Factor ${(1 - x + x^{2})}^{2}$ out of $3 {(1 - x + x^{2})}^{2} x^{5} x' (2 - x^{3})$ .

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

Step 3.18.4.5.2

Cancel the common factors.

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Step 3.18.4.5.2.1

Factor ${(1 - x + x^{2})}^{2}$ out of ${(1 + x)}^{4} {(1 - x + x^{2})}^{6}$ .

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

Step 3.18.4.5.2.2

Cancel the common factor.

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

Step 3.18.4.5.2.3

Rewrite the expression.

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

Step 3.18.5

Reorder terms.

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{3 x^{5} (2 - x^{3}) x'}{{(1 + x)}^{4} {(1 - x + x^{2})}^{4}} = 1$ .

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

Step 5.2

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

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

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

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Step 5.3.1.1.1

Simplify terms.

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Step 5.3.1.1.1.1

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

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Step 5.3.1.1.1.1.1

Cancel the common factor.

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

Step 5.3.1.1.1.1.2

Rewrite the expression.

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

Step 5.3.1.1.1.2

Apply the distributive property.

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

Step 5.3.1.1.1.3

Simplify the expression.

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Step 5.3.1.1.1.3.1

Multiply $2$ by $3$ .

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

Step 5.3.1.1.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.1.2

Simplify each term.

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Step 5.3.1.1.2.1

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 5.3.1.1.2.1.1

Move $x^{3}$ .

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

Step 5.3.1.1.2.1.2

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

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

Step 5.3.1.1.2.1.3

Add $3$ and $5$ .

$(6 x^{5} + 3 \cdot - 1 x^{8}) x' = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.1.1.2.2

Multiply $3$ by $- 1$ .

$(6 x^{5} - 3 x^{8}) x' = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.1.1.3

Simplify by multiplying through.

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Step 5.3.1.1.3.1

Apply the distributive property.

$6 x^{5} x' - 3 x^{8} x' = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.1.1.3.2

Simplify the expression.

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Step 5.3.1.1.3.2.1

Move $x^{5}$ .

$6 x' x^{5} - 3 x^{8} x' = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.1.1.3.2.2

Move $x^{8}$ .

$6 x' x^{5} - 3 x' x^{8} = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.1.1.3.2.3

Reorder $6 x' x^{5}$ and $- 3 x' x^{8}$ .

$- 3 x' x^{8} + 6 x' x^{5} = 1 ({(1 + x)}^{4} {(1 - x + x^{2})}^{4})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

$- 3 x' x^{8} + 6 x' x^{5} = {(1 + x)}^{4} {(1 - x + x^{2})}^{4}$

Step 5.4

Solve for $x'$ .

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Step 5.4.1

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

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Step 5.4.1.1

Use the Binomial Theorem.

$- 3 x' x^{8} + 6 x' x^{5} = (1^{4} + 4 \cdot 1^{3} x + 6 \cdot 1^{2} x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

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.

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 \cdot 1^{3} x + 6 \cdot 1^{2} x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.1.2

One to any power is one.

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 \cdot 1 x + 6 \cdot 1^{2} x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.1.3

Multiply $4$ by $1$ .

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 x + 6 \cdot 1^{2} x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.1.4

One to any power is one.

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 x + 6 \cdot 1 x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.1.5

Multiply $6$ by $1$ .

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 x + 6 x^{2} + 4 \cdot 1 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.1.6

Multiply $4$ by $1$ .

$- 3 x' x^{8} + 6 x' x^{5} = (1 + 4 x + 6 x^{2} + 4 x^{3} + x^{4}) {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.2

Simplify by multiplying through.

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Step 5.4.1.2.2.1

Apply the distributive property.

$- 3 x' x^{8} + 6 x' x^{5} = 1 {(1 - x + x^{2})}^{4} + 4 x {(1 - x + x^{2})}^{4} + 6 x^{2} {(1 - x + x^{2})}^{4} + 4 x^{3} {(1 - x + x^{2})}^{4} + x^{4} {(1 - x + x^{2})}^{4}$

Step 5.4.1.2.2.2

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

$- 3 x' x^{8} + 6 x' x^{5} = {(1 - x + x^{2})}^{4} + 4 x {(1 - x + x^{2})}^{4} + 6 x^{2} {(1 - x + x^{2})}^{4} + 4 x^{3} {(1 - x + x^{2})}^{4} + x^{4} {(1 - x + x^{2})}^{4}$

Step 5.4.2

Factor $3 x' x^{5}$ out of $- 3 x' x^{8} + 6 x' x^{5}$ .

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Step 5.4.2.1

Factor $3 x' x^{5}$ out of $- 3 x' x^{8}$ .

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

Step 5.4.2.2

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

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

Step 5.4.2.3

Factor $3 x' x^{5}$ out of $3 x' x^{5} (- 1 x^{3}) + 3 x' x^{5} \cdot 2$ .

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

Step 5.4.3

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

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

Step 5.4.4

Divide each term in $3 x' x^{5} (- x^{3} + 2) = {(1 - x + x^{2})}^{4} + 4 x {(1 - x + x^{2})}^{4} + 6 x^{2} {(1 - x + x^{2})}^{4} + 4 x^{3} {(1 - x + x^{2})}^{4} + x^{4} {(1 - x + x^{2})}^{4}$ by $3 x^{5} (- x^{3} + 2)$ and simplify.

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Step 5.4.4.1

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

Step 5.4.4.2

Simplify the left side.

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Step 5.4.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.4.2.1.1

Cancel the common factor.

Step 5.4.4.2.1.2

Rewrite the expression.

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

Step 5.4.4.2.2

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

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Step 5.4.4.2.2.1

Cancel the common factor.

Step 5.4.4.2.2.2

Rewrite the expression.

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

Step 5.4.4.2.3

Cancel the common factor of $- x^{3} + 2$ .

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Step 5.4.4.2.3.1

Cancel the common factor.

Step 5.4.4.2.3.2

Divide $x'$ by $1$ .

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

Step 5.4.4.3

Simplify the right side.

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Step 5.4.4.3.1

Combine the numerators over the common denominator.

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

Step 5.4.4.3.2

Simplify the numerator.

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Step 5.4.4.3.2.1

Factor ${(1 - x + x^{2})}^{4}$ out of ${(1 - x + x^{2})}^{4} + 4 x {(1 - x + x^{2})}^{4} + 6 x^{2} {(1 - x + x^{2})}^{4} + 4 x^{3} {(1 - x + x^{2})}^{4} + x^{4} {(1 - x + x^{2})}^{4}$ .

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Step 5.4.4.3.2.1.1

Multiply by $1$ .

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

Step 5.4.4.3.2.1.2

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

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

Step 5.4.4.3.2.1.3

Factor ${(1 - x + x^{2})}^{4}$ out of $6 x^{2} {(1 - x + x^{2})}^{4}$ .

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

Step 5.4.4.3.2.1.4

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

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

Step 5.4.4.3.2.1.5

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

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

Step 5.4.4.3.2.1.6

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

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

Step 5.4.4.3.2.1.7

Factor ${(1 - x + x^{2})}^{4}$ out of ${(1 - x + x^{2})}^{4} \cdot (1 + 4 x) + {(1 - x + x^{2})}^{4} (6 x^{2})$ .

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

Step 5.4.4.3.2.1.8

Factor ${(1 - x + x^{2})}^{4}$ out of ${(1 - x + x^{2})}^{4} \cdot (1 + 4 x + 6 x^{2}) + {(1 - x + x^{2})}^{4} (4 x^{3})$ .

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

Step 5.4.4.3.2.1.9

Factor ${(1 - x + x^{2})}^{4}$ out of ${(1 - x + x^{2})}^{4} \cdot (1 + 4 x + 6 x^{2} + 4 x^{3}) + {(1 - x + x^{2})}^{4} x^{4}$ .

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

Step 5.4.4.3.2.2

Move $1$ .

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

Step 5.4.4.3.2.3

Move $4 x$ .

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

Step 5.4.4.3.2.4

Move $6 x^{2}$ .

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

Step 5.4.4.3.2.5

Reorder $4 x^{3}$ and $x^{4}$ .

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

Step 5.4.4.3.2.6

Factor $x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1$ using the binomial theorem.

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

Step 5.4.4.3.3

Simplify with factoring out.

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Step 5.4.4.3.3.1

Factor $- 1$ out of $- x^{3}$ .

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

Step 5.4.4.3.3.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 5.4.4.3.3.3

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

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

Step 5.4.4.3.3.4

Rewrite negatives.

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Step 5.4.4.3.3.4.1

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

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

Step 5.4.4.3.3.4.2

Move the negative in front of the fraction.

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

Step 6

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

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