Calculus Examples

$(3 x y + 3 y - 4) d x + {(x + 1)}^{2} d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 x y + 3 y - 4$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x y + 3 y - 4]$

Step 1.2

By the Sum Rule, the derivative of $3 x y + 3 y - 4$ with respect to $y$ is $\frac{d}{d y} [3 x y] + \frac{d}{d y} [3 y] + \frac{d}{d y} [- 4]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x y] + \frac{d}{d y} [3 y] + \frac{d}{d y} [- 4]$

Step 1.3

Evaluate $\frac{d}{d y} [3 x y]$ .

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 3 x \frac{d}{d y} [y] + \frac{d}{d y} [3 y] + \frac{d}{d y} [- 4]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 3 x \cdot 1 + \frac{d}{d y} [3 y] + \frac{d}{d y} [- 4]$

Step 1.3.3

Multiply $3$ by $1$ .

$\frac{\partial M}{\partial y} = 3 x + \frac{d}{d y} [3 y] + \frac{d}{d y} [- 4]$

Step 1.4

Evaluate $\frac{d}{d y} [3 y]$ .

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Step 1.4.1

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

$\frac{\partial M}{\partial y} = 3 x + 3 \frac{d}{d y} [y] + \frac{d}{d y} [- 4]$

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 3 x + 3 \cdot 1 + \frac{d}{d y} [- 4]$

Step 1.4.3

Multiply $3$ by $1$ .

$\frac{\partial M}{\partial y} = 3 x + 3 + \frac{d}{d y} [- 4]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

Since $- 4$ is constant with respect to $y$ , the derivative of $- 4$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 3 x + 3 + 0$

Step 1.5.2

Add $3 x + 3$ and $0$ .

$\frac{\partial M}{\partial y} = 3 x + 3$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = {(x + 1)}^{2}$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [{(x + 1)}^{2}]$

Step 2.2

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

$\frac{\partial N}{\partial x} = \frac{d}{d x} [(x + 1) (x + 1)]$

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x (x + 1) + 1 (x + 1)]$

Step 2.3.2

Apply the distributive property.

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x \cdot x + x \cdot 1 + 1 (x + 1)]$

Step 2.3.3

Apply the distributive property.

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1]$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $x$ by $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + x \cdot 1 + 1 x + 1 \cdot 1]$

Step 2.4.1.2

Multiply $x$ by $1$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + x + 1 x + 1 \cdot 1]$

Step 2.4.1.3

Multiply $x$ by $1$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + x + x + 1 \cdot 1]$

Step 2.4.1.4

Multiply $1$ by $1$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + x + x + 1]$

Step 2.4.2

Add $x$ and $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + 2 x + 1]$

Step 2.5

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

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 2.6

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

$\frac{\partial N}{\partial x} = 2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 2.7

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

$\frac{\partial N}{\partial x} = 2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 2.8

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

$\frac{\partial N}{\partial x} = 2 x + 2 \cdot 1 + \frac{d}{d x} [1]$

Step 2.9

Multiply $2$ by $1$ .

$\frac{\partial N}{\partial x} = 2 x + 2 + \frac{d}{d x} [1]$

Step 2.10

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

$\frac{\partial N}{\partial x} = 2 x + 2 + 0$

Step 2.11

Add $2 x + 2$ and $0$ .

$\frac{\partial N}{\partial x} = 2 x + 2$

Step 3

Check that $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ .

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Step 3.1

Substitute $3 x + 3$ for $\frac{\partial M}{\partial y}$ and $2 x + 2$ for $\frac{\partial N}{\partial x}$ .

$3 x + 3 = 2 x + 2$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$3 x + 3 = 2 x + 2$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

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Step 4.1

Substitute $3 x + 3$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x + 3 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $2 x + 2$ for $\frac{\partial N}{\partial x}$ .

$\frac{3 x + 3 - (2 x + 2)}{N}$

Step 4.3

Substitute ${(x + 1)}^{2}$ for $N$ .

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Step 4.3.1

Substitute ${(x + 1)}^{2}$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2.3

Multiply $- 1$ by $2$ .

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

Step 4.3.2.4

Subtract $2 x$ from $3 x$ .

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

Step 4.3.2.5

Subtract $2$ from $3$ .

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

Step 4.3.3

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

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Step 4.3.3.1

Multiply by $1$ .

$\frac{(x + 1) \cdot 1}{{(x + 1)}^{2}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

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

$\frac{(x + 1) \cdot 1}{(x + 1) (x + 1)}$

Step 4.3.3.2.2

Cancel the common factor.

$\frac{(x + 1) \cdot 1}{(x + 1) (x + 1)}$

Step 4.3.3.2.3

Rewrite the expression.

$\frac{1}{x + 1}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int \frac{1}{x + 1} d x}$

Step 5

Evaluate the integral $e^{\int \frac{1}{x + 1} d x}$ .

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Step 5.1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 5.1.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 5.1.1.3

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

$1 + \frac{d}{d x} [1]$

Step 5.1.1.4

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

$1 + 0$

Step 5.1.1.5

Add $1$ and $0$ .

$1$

Step 5.1.2

Rewrite the problem using $u$ and $d u$ .

$μ (x, y) = e^{\int \frac{1}{u} d u}$

Step 5.2

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$μ (x, y) = e^{\ln (| u |) + C}$

Step 5.3

Simplify.

$μ (x, y) = e^{\ln (| u |)} + C$

Step 5.4

Exponentiation and log are inverse functions.

$μ (x, y) = | u | + C$

Step 5.5

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

$μ (x, y) = | x + 1 | + C$

Step 6

Multiply both sides of $(3 x y + 3 y - 4) d x + {(x + 1)}^{2} d y = 0$ by the integration factor $x + 1$ .

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Step 6.1

Multiply $3 x y + 3 y - 4$ by $x + 1$ .

$(3 x y + 3 y - 4) (x + 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.2

Expand $(3 x y + 3 y - 4) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$(3 x y x + 3 x y \cdot 1 + 3 y x + 3 y \cdot 1 - 4 x - 4 \cdot 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.3

Simplify each term.

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Step 6.3.1

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

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Step 6.3.1.1

Move $x$ .

$(3 (x \cdot x) y + 3 x y \cdot 1 + 3 y x + 3 y \cdot 1 - 4 x - 4 \cdot 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.3.1.2

Multiply $x$ by $x$ .

$(3 x^{2} y + 3 x y \cdot 1 + 3 y x + 3 y \cdot 1 - 4 x - 4 \cdot 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.3.2

Multiply $3$ by $1$ .

$(3 x^{2} y + 3 x y + 3 y x + 3 y \cdot 1 - 4 x - 4 \cdot 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.3.3

Multiply $3$ by $1$ .

$(3 x^{2} y + 3 x y + 3 y x + 3 y - 4 x - 4 \cdot 1) d x + {(x + 1)}^{2} d y = 0$

Step 6.3.4

Multiply $- 4$ by $1$ .

$(3 x^{2} y + 3 x y + 3 y x + 3 y - 4 x - 4) d x + {(x + 1)}^{2} d y = 0$

Step 6.4

Add $3 x y$ and $3 y x$ .

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Step 6.4.1

Move $y$ .

$(3 x^{2} y + 3 x y + 3 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{2} d y = 0$

Step 6.4.2

Add $3 x y$ and $3 x y$ .

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{2} d y = 0$

Step 6.5

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

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{2} (x + 1) d y = 0$

Step 6.6

Multiply ${(x + 1)}^{2}$ by $x + 1$ by adding the exponents.

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Step 6.6.1

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

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Step 6.6.1.1

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

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{2} {(x + 1)}^{1} d y = 0$

Step 6.6.1.2

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

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{2 + 1} d y = 0$

Step 6.6.2

Add $2$ and $1$ .

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + {(x + 1)}^{3} d y = 0$

Step 6.7

Use the Binomial Theorem.

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) d y = 0$

Step 6.8

Simplify each term.

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Step 6.8.1

Multiply $3$ by $1$ .

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) d y = 0$

Step 6.8.2

One to any power is one.

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) d y = 0$

Step 6.8.3

Multiply $3$ by $1$ .

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + (x^{3} + 3 x^{2} + 3 x + 1^{3}) d y = 0$

Step 6.8.4

One to any power is one.

$(3 x^{2} y + 6 x y + 3 y - 4 x - 4) d x + (x^{3} + 3 x^{2} + 3 x + 1) d y = 0$

Step 7

Set $f (x, y)$ equal to the integral of $N (x, y)$ .

$f (x, y) = \int x^{3} + 3 x^{2} + 3 x + 1 d y$

Step 8

Integrate $N (x, y) = x^{3} + 3 x^{2} + 3 x + 1$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = (x^{3} + 3 x^{2} + 3 x + 1) y + C$

Step 9

Since the integral of $g (x)$ will contain an integration constant, we can replace $C$ with $g (x)$ .

$f (x, y) = (x^{3} + 3 x^{2} + 3 x + 1) y + g (x)$

Step 10

Set $\frac{\partial f}{\partial x} = M (x, y)$ .

$\frac{\partial f}{\partial x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11

Find $\frac{\partial f}{\partial x}$ .

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

By the Sum Rule, the derivative of $(x^{3} + 3 x^{2} + 3 x + 1) y + g (x)$ with respect to $x$ is $\frac{d}{d x} [(x^{3} + 3 x^{2} + 3 x + 1) y] + \frac{d}{d x} [g (x)]$ .

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

Step 11.3

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

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Step 11.3.1

Since $y$ is constant with respect to $x$ , the derivative of $(x^{3} + 3 x^{2} + 3 x + 1) y$ with respect to $x$ is $y \frac{d}{d x} [x^{3} + 3 x^{2} + 3 x + 1]$ .

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

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

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

Step 11.3.8

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

$y (3 x^{2} + 3 (2 x) + 3 \cdot 1 + 0) + \frac{d}{d x} [g (x)] = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.3.9

Multiply $2$ by $3$ .

$y (3 x^{2} + 6 x + 3 \cdot 1 + 0) + \frac{d}{d x} [g (x)] = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.3.10

Multiply $3$ by $1$ .

$y (3 x^{2} + 6 x + 3 + 0) + \frac{d}{d x} [g (x)] = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.3.11

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

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

Step 11.4

Differentiate using the function rule which states that the derivative of $g (x)$ is $\frac{d g}{d x}$ .

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

Step 11.5

Simplify.

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Step 11.5.1

Apply the distributive property.

$y (3 x^{2}) + y (6 x) + y \cdot 3 + \frac{d g}{d x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.5.2

Combine terms.

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Step 11.5.2.1

Move $3$ to the left of $y$ .

$3 \cdot y x^{2} + y (6 x) + y \cdot 3 + \frac{d g}{d x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.5.2.2

Move $6$ to the left of $y$ .

$3 y x^{2} + 6 \cdot y x + y \cdot 3 + \frac{d g}{d x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.5.2.3

Move $3$ to the left of $y$ .

$3 y x^{2} + 6 y x + 3 y + \frac{d g}{d x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 11.5.3

Reorder terms.

$\frac{d g}{d x} + 3 x^{2} y + 6 x y + 3 y = 3 x^{2} y + 6 x y + 3 y - 4 x - 4$

Step 12

Solve for $\frac{d g}{d x}$ .

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Step 12.1

Move all terms not containing $\frac{d g}{d x}$ to the right side of the equation.

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Step 12.1.1

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

$\frac{d g}{d x} + 6 x y + 3 y = 3 x^{2} y + 6 x y + 3 y - 4 x - 4 - 3 x^{2} y$

Step 12.1.2

Subtract $6 x y$ from both sides of the equation.

$\frac{d g}{d x} + 3 y = 3 x^{2} y + 6 x y + 3 y - 4 x - 4 - 3 x^{2} y - 6 x y$

Step 12.1.3

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

$\frac{d g}{d x} = 3 x^{2} y + 6 x y + 3 y - 4 x - 4 - 3 x^{2} y - 6 x y - 3 y$

Step 12.1.4

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

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Step 12.1.4.1

Subtract $3 x^{2} y$ from $3 x^{2} y$ .

$\frac{d g}{d x} = 6 x y + 3 y - 4 x - 4 + 0 - 6 x y - 3 y$

Step 12.1.4.2

Add $6 x y + 3 y - 4 x - 4$ and $0$ .

$\frac{d g}{d x} = 6 x y + 3 y - 4 x - 4 - 6 x y - 3 y$

Step 12.1.4.3

Subtract $6 x y$ from $6 x y$ .

$\frac{d g}{d x} = 3 y - 4 x - 4 + 0 - 3 y$

Step 12.1.4.4

Add $3 y - 4 x - 4$ and $0$ .

$\frac{d g}{d x} = 3 y - 4 x - 4 - 3 y$

Step 12.1.4.5

Subtract $3 y$ from $3 y$ .

$\frac{d g}{d x} = - 4 x - 4 + 0$

Step 12.1.4.6

Add $- 4 x - 4$ and $0$ .

$\frac{d g}{d x} = - 4 x - 4$

Step 13

Find the antiderivative of $- 4 x - 4$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = - 4 x - 4$ .

$\int \frac{d g}{d x} d x = \int - 4 x - 4 d x$

Step 13.2

Evaluate $\int \frac{d g}{d x} d x$ .

$g (x) = \int - 4 x - 4 d x$

Step 13.3

Split the single integral into multiple integrals.

$g (x) = \int - 4 x d x + \int - 4 d x$

Step 13.4

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$g (x) = - 4 \int x d x + \int - 4 d x$

Step 13.5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$g (x) = - 4 (\frac{1}{2} x^{2} + C) + \int - 4 d x$

Step 13.6

Apply the constant rule.

$g (x) = - 4 (\frac{1}{2} x^{2} + C) - 4 x + C$

Step 13.7

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

$g (x) = - 4 (\frac{x^{2}}{2} + C) - 4 x + C$

Step 13.8

Simplify.

$g (x) = - 2 x^{2} - 4 x + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = (x^{3} + 3 x^{2} + 3 x + 1) y + g (x)$ .

$f (x, y) = (x^{3} + 3 x^{2} + 3 x + 1) y - 2 x^{2} - 4 x + C$

Step 15

Simplify each term.

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Step 15.1

Apply the distributive property.

$f (x, y) = x^{3} y + 3 x^{2} y + 3 x y + 1 y - 2 x^{2} - 4 x + C$

Step 15.2

Multiply $y$ by $1$ .

$f (x, y) = x^{3} y + 3 x^{2} y + 3 x y + y - 2 x^{2} - 4 x + C$