Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

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 $g (y) = 3 x - 1$ .

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

Differentiate.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.6

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

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

Step 3.7

Differentiate using the Constant Rule.

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Step 3.7.1

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

$\frac{2 {(x^{2} + 3)}^{2} (3 x - 1) (3 x' + 0) - {(3 x - 1)}^{2} \frac{d}{d y} [{(x^{2} + 3)}^{2}]}{{(x^{2} + 3)}^{4}}$

Step 3.7.2

Simplify the expression.

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Step 3.7.2.1

Add $3 x'$ and $0$ .

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

Step 3.7.2.2

Multiply $3$ by $2$ .

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

Step 3.8

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

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Step 3.8.1

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

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

Step 3.8.2

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

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

Step 3.8.3

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

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

Step 3.9

Differentiate using the Sum Rule.

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Step 3.9.1

Multiply $2$ by $- 1$ .

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

Step 3.9.2

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

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

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 $f (y) = y^{2}$ and $g (y) = x$ .

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Step 3.10.1

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

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

Step 3.10.2

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

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

Step 3.10.3

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

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

Step 3.11

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

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

Step 3.12

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) x' - 2 {(3 x - 1)}^{2} ((x^{2} + 3) (2 x x' + 0))}{{(x^{2} + 3)}^{4}}$

Step 3.13

Simplify the expression.

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Step 3.13.1

Add $2 x x'$ and $0$ .

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

Step 3.13.2

Move $2$ to the left of $x^{2} + 3$ .

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

Step 3.13.3

Multiply $2$ by $- 2$ .

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

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

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

Step 3.14.2

Apply the distributive property.

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

Step 3.14.3

Simplify the numerator.

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Step 3.14.3.1

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

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Step 3.14.3.1.1

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

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

Step 3.14.3.1.2

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

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

Step 3.14.3.1.3

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

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

Step 3.14.3.2

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

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Step 3.14.3.2.1

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

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Step 3.14.3.2.1.1

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

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

Step 3.14.3.2.1.2

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

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

Step 3.14.3.2.2

Add $2$ and $1$ .

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

Step 3.14.3.3

Simplify each term.

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Step 3.14.3.3.1

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

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

Step 3.14.3.3.2

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

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Step 3.14.3.3.2.1

Apply the distributive property.

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

Step 3.14.3.3.2.2

Apply the distributive property.

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

Step 3.14.3.3.2.3

Apply the distributive property.

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

Step 3.14.3.3.3

Simplify and combine like terms.

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Step 3.14.3.3.3.1

Simplify each term.

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Step 3.14.3.3.3.1.1

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

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Step 3.14.3.3.3.1.1.1

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

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

Step 3.14.3.3.3.1.1.2

Add $2$ and $2$ .

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

Step 3.14.3.3.3.1.2

Move $3$ to the left of $x^{2}$ .

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

Step 3.14.3.3.3.1.3

Multiply $3$ by $3$ .

$\frac{2 (3 x - 1) (3 (x^{4} + 3 x^{2} + 3 x^{2} + 9) x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.3.2

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

$\frac{2 (3 x - 1) (3 (x^{4} + 6 x^{2} + 9) x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.4

Apply the distributive property.

$\frac{2 (3 x - 1) ((3 x^{4} + 3 (6 x^{2}) + 3 \cdot 9) x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.5

Simplify.

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Step 3.14.3.3.5.1

Multiply $6$ by $3$ .

$\frac{2 (3 x - 1) ((3 x^{4} + 18 x^{2} + 3 \cdot 9) x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.5.2

Multiply $3$ by $9$ .

$\frac{2 (3 x - 1) ((3 x^{4} + 18 x^{2} + 27) x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.6

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 2 (3 x - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.7

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' + (- 2 (3 x) - 2 \cdot - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.8

Multiply $3$ by $- 2$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' + (- 6 x - 2 \cdot - 1) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.9

Multiply $- 2$ by $- 1$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' + (- 6 x + 2) (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.10

Expand $(- 6 x + 2) (x^{3} x' + 3 x x')$ using the FOIL Method.

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Step 3.14.3.3.10.1

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x (x^{3} x' + 3 x x') + 2 (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.10.2

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x (x^{3} x') - 6 x (3 x x') + 2 (x^{3} x' + 3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.10.3

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x (x^{3} x') - 6 x (3 x x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11

Simplify each term.

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Step 3.14.3.3.11.1

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

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Step 3.14.3.3.11.1.1

Move $x^{3}$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 (x^{3} x) x' - 6 x (3 x x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.1.2

Multiply $x^{3}$ by $x$ .

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Step 3.14.3.3.11.1.2.1

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

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 (x^{3} x^{1}) x' - 6 x (3 x x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.1.2.2

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

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{3 + 1} x' - 6 x (3 x x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.1.3

Add $3$ and $1$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 6 x (3 x x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.2

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

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Step 3.14.3.3.11.2.1

Move $x$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 6 (x \cdot x) (3 x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.2.2

Multiply $x$ by $x$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 6 x^{2} (3 x') + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.3

Multiply $3$ by $- 6$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 18 x^{2} x' + 2 (x^{3} x') + 2 (3 x x'))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.3.11.4

Multiply $3$ by $2$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 18 x^{2} x' + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.4

Combine the opposite terms in $3 x^{4} x' + 18 x^{2} x' + 27 x' - 6 x^{4} x' - 18 x^{2} x' + 2 x^{3} x' + 6 x x'$ .

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Step 3.14.3.4.1

Subtract $18 x^{2} x'$ from $18 x^{2} x'$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 27 x' - 6 x^{4} x' + 0 + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.4.2

Add $3 x^{4} x' + 27 x' - 6 x^{4} x'$ and $0$ .

$\frac{2 (3 x - 1) (3 x^{4} x' + 27 x' - 6 x^{4} x' + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.5

Subtract $6 x^{4} x'$ from $3 x^{4} x'$ .

$\frac{2 (3 x - 1) (- 3 x^{4} x' + 27 x' + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6

Factor $x'$ out of $- 3 x^{4} x' + 27 x' + 2 x^{3} x' + 6 x x'$ .

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Step 3.14.3.6.1

Factor $x'$ out of $- 3 x^{4} x'$ .

$\frac{2 (3 x - 1) (x' (- 3 x^{4}) + 27 x' + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.2

Factor $x'$ out of $27 x'$ .

$\frac{2 (3 x - 1) (x' (- 3 x^{4}) + x' \cdot 27 + 2 x^{3} x' + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.3

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

$\frac{2 (3 x - 1) (x' (- 3 x^{4}) + x' \cdot 27 + x' (2 x^{3}) + 6 x x')}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.4

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

$\frac{2 (3 x - 1) (x' (- 3 x^{4}) + x' \cdot 27 + x' (2 x^{3}) + x' (6 x))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.5

Factor $x'$ out of $x' (- 3 x^{4}) + x' \cdot 27$ .

$\frac{2 (3 x - 1) (x' (- 3 x^{4} + 27) + x' (2 x^{3}) + x' (6 x))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.6

Factor $x'$ out of $x' (- 3 x^{4} + 27) + x' (2 x^{3})$ .

$\frac{2 (3 x - 1) (x' (- 3 x^{4} + 27 + 2 x^{3}) + x' (6 x))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.6.7

Factor $x'$ out of $x' (- 3 x^{4} + 27 + 2 x^{3}) + x' (6 x)$ .

$\frac{2 (3 x - 1) (x' (- 3 x^{4} + 27 + 2 x^{3} + 6 x))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.7

Reorder terms.

$\frac{2 (3 x - 1) (x' (- 3 x^{4} + 2 x^{3} + 6 x + 27))}{{(x^{2} + 3)}^{4}}$

Step 3.14.3.8

Factor.

$\frac{2 (3 x - 1) x' (x^{2} + 3) (- 3 x^{2} + 2 x + 9)}{{(x^{2} + 3)}^{4}}$

Step 3.14.4

Cancel the common factor of $x^{2} + 3$ and ${(x^{2} + 3)}^{4}$ .

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Step 3.14.4.1

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

$\frac{(x^{2} + 3) (2 (3 x - 1) x' (- 3 x^{2} + 2 x + 9))}{{(x^{2} + 3)}^{4}}$

Step 3.14.4.2

Cancel the common factors.

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Step 3.14.4.2.1

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

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

Step 3.14.4.2.2

Cancel the common factor.

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

Step 3.14.4.2.3

Rewrite the expression.

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

Step 3.14.5

Reorder terms.

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

Step 3.14.6

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

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

Step 3.14.7

Factor $- 1$ out of $2 x$ .

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

Step 3.14.8

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

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

Step 3.14.9

Rewrite $9$ as $- 1 (- 9)$ .

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

Step 3.14.10

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

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

Step 3.14.11

Rewrite $- (3 x^{2} - 2 x - 9)$ as $- 1 (3 x^{2} - 2 x - 9)$ .

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

Step 3.14.12

Move the negative in front of the fraction.

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

Step 3.14.13

Reorder factors in $- \frac{((2 (3 x^{2} - 2 x - 9)) (3 x - 1)) x'}{{(x^{2} + 3)}^{3}}$ .

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{2 (3 x^{2} - 2 x - 9) (3 x - 1) x'}{{(x^{2} + 3)}^{3}} = 1$ .

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

Step 5.2

Divide each term in $- \frac{2 (3 x^{2} - 2 x - 9) (3 x - 1) x'}{{(x^{2} + 3)}^{3}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{2 (3 x^{2} - 2 x - 9) (3 x - 1) x'}{{(x^{2} + 3)}^{3}} = 1$ by $- 1$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

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

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

Step 5.2.2.2.2

Reorder factors in $\frac{2 (3 x^{2} - 2 x - 9) (3 x - 1) x'}{{(x^{2} + 3)}^{3}}$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

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

Step 5.3

Multiply both sides by ${(x^{2} + 3)}^{3}$ .

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

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Cancel the common factor.

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

Step 5.4.1.1.2

Rewrite the expression.

$2 x' (3 x^{2} - 2 x - 9) (3 x - 1) = - {(x^{2} + 3)}^{3}$

Step 5.4.1.2

Apply the distributive property.

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

Step 5.4.1.3

Simplify.

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Step 5.4.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 5.4.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 5.4.1.3.3

Multiply $- 9$ by $2$ .

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

Step 5.4.1.4

Simplify each term.

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Step 5.4.1.4.1

Multiply $2$ by $3$ .

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

Step 5.4.1.4.2

Multiply $2$ by $- 2$ .

$(6 x' x^{2} - 4 x' x - 18 x') (3 x - 1) = - {(x^{2} + 3)}^{3}$

Step 5.4.1.5

Expand $(6 x' x^{2} - 4 x' x - 18 x') (3 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$6 x' x^{2} (3 x) + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6

Simplify terms.

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Step 5.4.1.6.1

Simplify each term.

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Step 5.4.1.6.1.1

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

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Step 5.4.1.6.1.1.1

Move $x$ .

$6 x' (x \cdot x^{2}) \cdot 3 + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.1.2

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

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Step 5.4.1.6.1.1.2.1

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

$6 x' (x^{1} x^{2}) \cdot 3 + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.1.2.2

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

$6 x' x^{1 + 2} \cdot 3 + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.1.3

Add $1$ and $2$ .

$6 x' x^{3} \cdot 3 + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.2

Multiply $3$ by $6$ .

$18 x' x^{3} + 6 x' x^{2} \cdot - 1 - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.3

Multiply $- 1$ by $6$ .

$18 x' x^{3} - 6 x' x^{2} - 4 x' x (3 x) - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.4

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

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Step 5.4.1.6.1.4.1

Move $x$ .

$18 x' x^{3} - 6 x' x^{2} - 4 x' (x \cdot x) \cdot 3 - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.4.2

Multiply $x$ by $x$ .

$18 x' x^{3} - 6 x' x^{2} - 4 x' x^{2} \cdot 3 - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.5

Multiply $3$ by $- 4$ .

$18 x' x^{3} - 6 x' x^{2} - 12 x' x^{2} - 4 x' x \cdot - 1 - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.6

Multiply $- 1$ by $- 4$ .

$18 x' x^{3} - 6 x' x^{2} - 12 x' x^{2} + 4 x' x - 18 x' (3 x) - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.7

Rewrite using the commutative property of multiplication.

$18 x' x^{3} - 6 x' x^{2} - 12 x' x^{2} + 4 x' x - 18 \cdot 3 x' x - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.8

Multiply $- 18$ by $3$ .

$18 x' x^{3} - 6 x' x^{2} - 12 x' x^{2} + 4 x' x - 54 x' x - 18 x' \cdot - 1 = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.1.9

Multiply $- 1$ by $- 18$ .

$18 x' x^{3} - 6 x' x^{2} - 12 x' x^{2} + 4 x' x - 54 x' x + 18 x' = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.2

Simplify by adding terms.

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Step 5.4.1.6.2.1

Subtract $12 x' x^{2}$ from $- 6 x' x^{2}$ .

$18 x' x^{3} - 18 x' x^{2} + 4 x' x - 54 x' x + 18 x' = - {(x^{2} + 3)}^{3}$

Step 5.4.1.6.2.2

Subtract $54 x' x$ from $4 x' x$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - {(x^{2} + 3)}^{3}$

Step 5.5

Solve for $x'$ .

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Step 5.5.1

Simplify $- {(x^{2} + 3)}^{3}$ .

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Step 5.5.1.1

Use the Binomial Theorem.

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - ({(x^{2})}^{3} + 3 {(x^{2})}^{2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2

Simplify terms.

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Step 5.5.1.2.1

Simplify each term.

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Step 5.5.1.2.1.1

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

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Step 5.5.1.2.1.1.1

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{2 \cdot 3} + 3 {(x^{2})}^{2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2.1.1.2

Multiply $2$ by $3$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 3 {(x^{2})}^{2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2.1.2

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

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Step 5.5.1.2.1.2.1

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 3 x^{2 \cdot 2} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2.1.2.2

Multiply $2$ by $2$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 3 x^{4} \cdot 3 + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2.1.3

Multiply $3$ by $3$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 3 x^{2} \cdot 3^{2} + 3^{3})$

Step 5.5.1.2.1.4

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 5.5.1.2.1.4.1

Move $3^{2}$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 3^{2} \cdot 3 x^{2} + 3^{3})$

Step 5.5.1.2.1.4.2

Multiply $3^{2}$ by $3$ .

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Step 5.5.1.2.1.4.2.1

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 3^{2} \cdot 3^{1} x^{2} + 3^{3})$

Step 5.5.1.2.1.4.2.2

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 3^{2 + 1} x^{2} + 3^{3})$

Step 5.5.1.2.1.4.3

Add $2$ and $1$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 3^{3} x^{2} + 3^{3})$

Step 5.5.1.2.1.5

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 27 x^{2} + 3^{3})$

Step 5.5.1.2.1.6

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

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - (x^{6} + 9 x^{4} + 27 x^{2} + 27)$

Step 5.5.1.2.2

Apply the distributive property.

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - x^{6} - (9 x^{4}) - (27 x^{2}) - 1 \cdot 27$

Step 5.5.1.3

Simplify.

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Step 5.5.1.3.1

Multiply $9$ by $- 1$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - x^{6} - 9 x^{4} - (27 x^{2}) - 1 \cdot 27$

Step 5.5.1.3.2

Multiply $27$ by $- 1$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - x^{6} - 9 x^{4} - 27 x^{2} - 1 \cdot 27$

Step 5.5.1.3.3

Multiply $- 1$ by $27$ .

$18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x' = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2

Factor $2 x'$ out of $18 x' x^{3} - 18 x' x^{2} - 50 x' x + 18 x'$ .

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Step 5.5.2.1

Factor $2 x'$ out of $18 x' x^{3}$ .

$2 x' (9 x^{3}) - 18 x' x^{2} - 50 x' x + 18 x' = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.2

Factor $2 x'$ out of $- 18 x' x^{2}$ .

$2 x' (9 x^{3}) + 2 x' (- 9 x^{2}) - 50 x' x + 18 x' = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.3

Factor $2 x'$ out of $- 50 x' x$ .

$2 x' (9 x^{3}) + 2 x' (- 9 x^{2}) + 2 x' (- 25 x) + 18 x' = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.4

Factor $2 x'$ out of $18 x'$ .

$2 x' (9 x^{3}) + 2 x' (- 9 x^{2}) + 2 x' (- 25 x) + 2 x' \cdot 9 = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.5

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

$2 x' (9 x^{3} - 9 x^{2}) + 2 x' (- 25 x) + 2 x' \cdot 9 = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.6

Factor $2 x'$ out of $2 x' (9 x^{3} - 9 x^{2}) + 2 x' (- 25 x)$ .

$2 x' (9 x^{3} - 9 x^{2} - 25 x) + 2 x' \cdot 9 = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.2.7

Factor $2 x'$ out of $2 x' (9 x^{3} - 9 x^{2} - 25 x) + 2 x' \cdot 9$ .

$2 x' (9 x^{3} - 9 x^{2} - 25 x + 9) = - x^{6} - 9 x^{4} - 27 x^{2} - 27$

Step 5.5.3

Divide each term in $2 x' (9 x^{3} - 9 x^{2} - 25 x + 9) = - x^{6} - 9 x^{4} - 27 x^{2} - 27$ by $2 (9 x^{3} - 9 x^{2} - 25 x + 9)$ and simplify.

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Step 5.5.3.1

Divide each term in $2 x' (9 x^{3} - 9 x^{2} - 25 x + 9) = - x^{6} - 9 x^{4} - 27 x^{2} - 27$ by $2 (9 x^{3} - 9 x^{2} - 25 x + 9)$ .

$\frac{2 x' (9 x^{3} - 9 x^{2} - 25 x + 9)}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} = \frac{- x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.2

Simplify the left side.

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Step 5.5.3.2.1

Cancel the common factor of $2$ .

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Step 5.5.3.2.1.1

Cancel the common factor.

Step 5.5.3.2.1.2

Rewrite the expression.

$\frac{x' (9 x^{3} - 9 x^{2} - 25 x + 9)}{9 x^{3} - 9 x^{2} - 25 x + 9} = \frac{- x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.2.2

Cancel the common factor of $9 x^{3} - 9 x^{2} - 25 x + 9$ .

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Step 5.5.3.2.2.1

Cancel the common factor.

Step 5.5.3.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.3

Simplify the right side.

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Step 5.5.3.3.1

Simplify each term.

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Step 5.5.3.3.1.1

Move the negative in front of the fraction.

$x' = - \frac{x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{(9) x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.3.1.3

Move the negative in front of the fraction.

$x' = - \frac{x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{(27) x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} + \frac{- 27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 5.5.3.3.1.4

Move the negative in front of the fraction.

$x' = - \frac{x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$

Step 6

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

$\frac{d x}{d y} = - \frac{x^{6}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{9 x^{4}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{27 x^{2}}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)} - \frac{27}{2 (9 x^{3} - 9 x^{2} - 25 x + 9)}$