Calculus Examples

${(x y + 1)}^{3} = x - y^{2} + 8$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x y + 1)}^{3}) = \frac{d}{d x} (x - y^{2} + 8)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x y + 1$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [x y + 1]$

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $x y + 1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $y$ by $1$ .

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

Step 2.7

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

$3 {(x y + 1)}^{2} (x y' + y + 0)$

Step 2.8

Add $x y' + y$ and $0$ .

$3 {(x y + 1)}^{2} (x y' + y)$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x - y^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [8]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [8]$

Step 3.2

Evaluate $\frac{d}{d x} [- y^{2}]$ .

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Step 3.2.1

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

$1 - \frac{d}{d x} [y^{2}] + \frac{d}{d x} [8]$

Step 3.2.2

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

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Step 3.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

$1 - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [8]$

Step 3.2.2.2

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

$1 - (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [8]$

Step 3.2.2.3

Replace all occurrences of $u$ with $y$ .

$1 - (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [8]$

Step 3.2.3

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

$1 - (2 y y') + \frac{d}{d x} [8]$

Step 3.2.4

Multiply $2$ by $- 1$ .

$1 - 2 y y' + \frac{d}{d x} [8]$

Step 3.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$1 - 2 y y' + 0$

Step 3.4

Simplify.

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Step 3.4.1

Add $1 - 2 y y'$ and $0$ .

$1 - 2 y y'$

Step 3.4.2

Reorder terms.

$- 2 y y' + 1$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $3 {(x y + 1)}^{2} (x y' + y)$ .

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Step 5.1.1

Rewrite.

$0 + 0 + 3 {(x y + 1)}^{2} (x y' + y) = - 2 y y' + 1$

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Reorder factors in $3 {(x y + 1)}^{2} (x y') + 3 {(x y + 1)}^{2} y$ .

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

Step 5.2

Move all terms containing $y'$ to the left side of the equation.

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Step 5.2.1

Add $2 y y'$ to both sides of the equation.

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Rewrite ${(x y + 1)}^{2}$ as $(x y + 1) (x y + 1)$ .

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

Step 5.2.2.2

Expand $(x y + 1) (x y + 1)$ using the FOIL Method.

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Step 5.2.2.2.1

Apply the distributive property.

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

Step 5.2.2.2.2

Apply the distributive property.

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

Step 5.2.2.2.3

Apply the distributive property.

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

Step 5.2.2.3

Simplify and combine like terms.

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Step 5.2.2.3.1

Simplify each term.

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Step 5.2.2.3.1.1

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

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Step 5.2.2.3.1.1.1

Move $x$ .

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

Step 5.2.2.3.1.1.2

Multiply $x$ by $x$ .

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

Step 5.2.2.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.3.1.2.1

Move $y$ .

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

Step 5.2.2.3.1.2.2

Multiply $y$ by $y$ .

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

Step 5.2.2.3.1.3

Multiply $x$ by $1$ .

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

Step 5.2.2.3.1.4

Multiply $x y$ by $1$ .

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

Step 5.2.2.3.1.5

Multiply $1$ by $1$ .

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

Step 5.2.2.3.2

Add $x y$ and $x y$ .

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

Step 5.2.2.4

Apply the distributive property.

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

Step 5.2.2.5

Simplify.

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Step 5.2.2.5.1

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

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Step 5.2.2.5.1.1

Move $x^{2}$ .

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

Step 5.2.2.5.1.2

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

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Step 5.2.2.5.1.2.1

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

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

Step 5.2.2.5.1.2.2

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

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

Step 5.2.2.5.1.3

Add $2$ and $1$ .

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

Step 5.2.2.5.2

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

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Step 5.2.2.5.2.1

Move $x$ .

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

Step 5.2.2.5.2.2

Multiply $x$ by $x$ .

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

Step 5.2.2.5.3

Multiply $3$ by $1$ .

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

Step 5.2.2.6

Multiply $2$ by $3$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y {(x y + 1)}^{2} + 2 y y' = 1$

Step 5.2.2.7

Rewrite ${(x y + 1)}^{2}$ as $(x y + 1) (x y + 1)$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y ((x y + 1) (x y + 1)) + 2 y y' = 1$

Step 5.2.2.8

Expand $(x y + 1) (x y + 1)$ using the FOIL Method.

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Step 5.2.2.8.1

Apply the distributive property.

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y (x y (x y + 1) + 1 (x y + 1)) + 2 y y' = 1$

Step 5.2.2.8.2

Apply the distributive property.

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

Step 5.2.2.8.3

Apply the distributive property.

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

Step 5.2.2.9

Simplify and combine like terms.

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Step 5.2.2.9.1

Simplify each term.

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Step 5.2.2.9.1.1

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

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Step 5.2.2.9.1.1.1

Move $x$ .

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

Step 5.2.2.9.1.1.2

Multiply $x$ by $x$ .

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

Step 5.2.2.9.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.9.1.2.1

Move $y$ .

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

Step 5.2.2.9.1.2.2

Multiply $y$ by $y$ .

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

Step 5.2.2.9.1.3

Multiply $x$ by $1$ .

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

Step 5.2.2.9.1.4

Multiply $x y$ by $1$ .

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

Step 5.2.2.9.1.5

Multiply $1$ by $1$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y (x^{2} y^{2} + x y + x y + 1) + 2 y y' = 1$

Step 5.2.2.9.2

Add $x y$ and $x y$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y (x^{2} y^{2} + 2 x y + 1) + 2 y y' = 1$

Step 5.2.2.10

Apply the distributive property.

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

Step 5.2.2.11

Simplify.

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Step 5.2.2.11.1

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

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Step 5.2.2.11.1.1

Move $y^{2}$ .

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

Step 5.2.2.11.1.2

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

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Step 5.2.2.11.1.2.1

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

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

Step 5.2.2.11.1.2.2

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

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

Step 5.2.2.11.1.3

Add $2$ and $1$ .

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

Step 5.2.2.11.2

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.11.2.1

Move $y$ .

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

Step 5.2.2.11.2.2

Multiply $y$ by $y$ .

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

Step 5.2.2.11.3

Multiply $3$ by $1$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y^{3} x^{2} + 3 y^{2} (2 x) + 3 y + 2 y y' = 1$

Step 5.2.2.12

Simplify each term.

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Step 5.2.2.12.1

Rewrite using the commutative property of multiplication.

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y^{3} x^{2} + 3 \cdot 2 y^{2} x + 3 y + 2 y y' = 1$

Step 5.2.2.12.2

Multiply $3$ by $2$ .

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y^{3} x^{2} + 6 y^{2} x + 3 y + 2 y y' = 1$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

Subtract $3 y^{3} x^{2}$ from both sides of the equation.

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 6 y^{2} x + 3 y + 2 y y' = 1 - 3 y^{3} x^{2}$

Step 5.3.2

Subtract $6 y^{2} x$ from both sides of the equation.

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 3 y + 2 y y' = 1 - 3 y^{3} x^{2} - 6 y^{2} x$

Step 5.3.3

Subtract $3 y$ from both sides of the equation.

$3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 2 y y' = 1 - 3 y^{3} x^{2} - 6 y^{2} x - 3 y$

Step 5.4

Factor $y'$ out of $3 x^{3} y' y^{2} + 6 x^{2} y' y + 3 x y' + 2 y y'$ .

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Step 5.4.1

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

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

Step 5.4.2

Factor $y'$ out of $6 x^{2} y' y$ .

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

Step 5.4.3

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

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

Step 5.4.4

Factor $y'$ out of $2 y y'$ .

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

Step 5.4.5

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

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

Step 5.4.6

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

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

Step 5.4.7

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

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $3 x^{3} y^{2} + 6 x^{2} y + 3 x + 2 y$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Divide $y'$ by $1$ .

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

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

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

Step 5.5.3.1.2

Move the negative in front of the fraction.

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

Step 5.5.3.1.3

Move the negative in front of the fraction.

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

Step 5.5.3.2

Combine into one fraction.

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Step 5.5.3.2.1

Combine the numerators over the common denominator.

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

Step 5.5.3.2.2

Combine the numerators over the common denominator.

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

Step 5.5.3.2.3

Combine the numerators over the common denominator.

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

Step 6

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

$\frac{d y}{d x} = \frac{1 - 3 y^{3} x^{2} - 6 y^{2} x - 3 y}{3 x^{3} y^{2} + 6 x^{2} y + 3 x + 2 y}$