Calculus Examples

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

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 2 (2 x y + 4 y - 3)$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 (2 x y + 4 y - 3)]$

Step 1.2

Since $2$ is constant with respect to $y$ , the derivative of $2 (2 x y + 4 y - 3)$ with respect to $y$ is $2 \frac{d}{d y} [2 x y + 4 y - 3]$ .

$\frac{\partial M}{\partial y} = 2 \frac{d}{d y} [2 x y + 4 y - 3]$

Step 1.3

By the Sum Rule, the derivative of $2 x y + 4 y - 3$ with respect to $y$ is $\frac{d}{d y} [2 x y] + \frac{d}{d y} [4 y] + \frac{d}{d y} [- 3]$ .

$\frac{\partial M}{\partial y} = 2 (\frac{d}{d y} [2 x y] + \frac{d}{d y} [4 y] + \frac{d}{d y} [- 3])$

Step 1.4

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

$\frac{\partial M}{\partial y} = 2 (2 x \frac{d}{d y} [y] + \frac{d}{d y} [4 y] + \frac{d}{d y} [- 3])$

Step 1.5

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

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

Step 1.6

Multiply $2$ by $1$ .

$\frac{\partial M}{\partial y} = 2 (2 x + \frac{d}{d y} [4 y] + \frac{d}{d y} [- 3])$

Step 1.7

Since $4$ is constant with respect to $y$ , the derivative of $4 y$ with respect to $y$ is $4 \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 2 (2 x + 4 \frac{d}{d y} [y] + \frac{d}{d y} [- 3])$

Step 1.8

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

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

Step 1.9

Multiply $4$ by $1$ .

$\frac{\partial M}{\partial y} = 2 (2 x + 4 + \frac{d}{d y} [- 3])$

Step 1.10

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

$\frac{\partial M}{\partial y} = 2 (2 x + 4 + 0)$

Step 1.11

Add $2 x + 4$ and $0$ .

$\frac{\partial M}{\partial y} = 2 (2 x + 4)$

Step 1.12

Simplify.

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Step 1.12.1

Apply the distributive property.

$\frac{\partial M}{\partial y} = 2 (2 x) + 2 \cdot 4$

Step 1.12.2

Combine terms.

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Step 1.12.2.1

Multiply $2$ by $2$ .

$\frac{\partial M}{\partial y} = 4 x + 2 \cdot 4$

Step 1.12.2.2

Multiply $2$ by $4$ .

$\frac{\partial M}{\partial y} = 4 x + 8$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

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 $x$ .

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

Step 2.4.1.2

Move $2$ to the left of $x$ .

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

Step 2.4.1.3

Multiply $2$ by $2$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + 2 x + 2 x + 4]$

Step 2.4.2

Add $2 x$ and $2 x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} + 4 x + 4]$

Step 2.5

By the Sum Rule, the derivative of $x^{2} + 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 2.6

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

$\frac{\partial N}{\partial x} = 2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 2.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [4]$

Step 2.8

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

$\frac{\partial N}{\partial x} = 2 x + 4 \cdot 1 + \frac{d}{d x} [4]$

Step 2.9

Multiply $4$ by $1$ .

$\frac{\partial N}{\partial x} = 2 x + 4 + \frac{d}{d x} [4]$

Step 2.10

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

$\frac{\partial N}{\partial x} = 2 x + 4 + 0$

Step 2.11

Add $2 x + 4$ and $0$ .

$\frac{\partial N}{\partial x} = 2 x + 4$

Step 3

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

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Step 3.1

Substitute $4 x + 8$ for $\frac{\partial M}{\partial y}$ and $2 x + 4$ for $\frac{\partial N}{\partial x}$ .

$4 x + 8 = 2 x + 4$

Step 3.2

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

$4 x + 8 = 2 x + 4$ 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 $4 x + 8$ for $\frac{\partial M}{\partial y}$ .

$\frac{4 x + 8 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $2 x + 4$ for $\frac{\partial N}{\partial x}$ .

$\frac{4 x + 8 - (2 x + 4)}{N}$

Step 4.3

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

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Step 4.3.1

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

$\frac{4 x + 8 - (2 x + 4)}{{(x + 2)}^{2}}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{4 x + 8 - (2 x) - 1 \cdot 4}{{(x + 2)}^{2}}$

Step 4.3.2.2

Multiply $2$ by $- 1$ .

$\frac{4 x + 8 - 2 x - 1 \cdot 4}{{(x + 2)}^{2}}$

Step 4.3.2.3

Multiply $- 1$ by $4$ .

$\frac{4 x + 8 - 2 x - 4}{{(x + 2)}^{2}}$

Step 4.3.2.4

Subtract $2 x$ from $4 x$ .

$\frac{2 x + 8 - 4}{{(x + 2)}^{2}}$

Step 4.3.2.5

Subtract $4$ from $8$ .

$\frac{2 x + 4}{{(x + 2)}^{2}}$

Step 4.3.2.6

Factor $2$ out of $2 x + 4$ .

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Step 4.3.2.6.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 4}{{(x + 2)}^{2}}$

Step 4.3.2.6.2

Factor $2$ out of $4$ .

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

Step 4.3.2.6.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

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

Step 4.3.3

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

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Step 4.3.3.1

Factor $x + 2$ out of $2 (x + 2)$ .

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

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

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

$\frac{2}{x + 2}$

Step 4.4

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

$μ (x, y) = e^{\int \frac{2}{x + 2} d x}$

Step 5

Evaluate the integral $e^{\int \frac{2}{x + 2} d x}$ .

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Step 5.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$μ (x, y) = e^{2 \int \frac{1}{x + 2} d x}$

Step 5.2

Let $u = x + 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.2.1

Let $u = x + 2$ . Find $\frac{d u}{d x}$ .

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Step 5.2.1.1

Differentiate $x + 2$ .

$\frac{d}{d x} [x + 2]$

Step 5.2.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 5.2.1.3

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

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

Step 5.2.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.2.1.5

Add $1$ and $0$ .

$1$

Step 5.2.2

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

$μ (x, y) = e^{2 \int \frac{1}{u} d u}$

Step 5.3

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

$μ (x, y) = e^{2 (\ln (| u |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{2 \ln (| u |)} + C$

Step 5.5

Replace all occurrences of $u$ with $x + 2$ .

$μ (x, y) = e^{2 \ln (| x + 2 |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $2 \ln (| x + 2 |)$ by moving $2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x + 2 |}^{2})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x + 2 |}^{2} + C$

Step 5.6.3

Remove the absolute value in ${| x + 2 |}^{2}$ because exponentiations with even powers are always positive.

$μ (x, y) = {(x + 2)}^{2} + C$

Step 5.6.4

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

$μ (x, y) = (x + 2) (x + 2) + C$

Step 5.6.5

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

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Step 5.6.5.1

Apply the distributive property.

$μ (x, y) = x (x + 2) + 2 (x + 2) + C$

Step 5.6.5.2

Apply the distributive property.

$μ (x, y) = x \cdot x + x \cdot 2 + 2 (x + 2) + C$

Step 5.6.5.3

Apply the distributive property.

$μ (x, y) = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2 + C$

Step 5.6.6

Simplify and combine like terms.

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Step 5.6.6.1

Simplify each term.

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Step 5.6.6.1.1

Multiply $x$ by $x$ .

$μ (x, y) = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2 + C$

Step 5.6.6.1.2

Move $2$ to the left of $x$ .

$μ (x, y) = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2 + C$

Step 5.6.6.1.3

Multiply $2$ by $2$ .

$μ (x, y) = x^{2} + 2 x + 2 x + 4 + C$

Step 5.6.6.2

Add $2 x$ and $2 x$ .

$μ (x, y) = x^{2} + 4 x + 4 + C$

Step 6

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

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Step 6.1

Multiply $2 (2 x y + 4 y - 3)$ by $x^{2} + 4 x + 4$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $2$ by $2$ .

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

Step 6.3.2

Multiply $4$ by $2$ .

$(4 (x y) + 8 y + 2 \cdot - 3) (x^{2} + 4 x + 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.3.3

Multiply $2$ by $- 3$ .

$(4 (x y) + 8 y - 6) (x^{2} + 4 x + 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.4

Expand $(4 x y + 8 y - 6) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(4 x y x^{2} + 4 x y (4 x) + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5

Simplify each term.

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Step 6.5.1

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

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Step 6.5.1.1

Move $x^{2}$ .

$(4 (x^{2} x) y + 4 x y (4 x) + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.1.2

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

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Step 6.5.1.2.1

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

$(4 (x^{2} x^{1}) y + 4 x y (4 x) + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.1.2.2

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

$(4 x^{2 + 1} y + 4 x y (4 x) + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.1.3

Add $2$ and $1$ .

$(4 x^{3} y + 4 x y (4 x) + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.2

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

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Step 6.5.2.1

Move $x$ .

$(4 x^{3} y + 4 (x \cdot x) y \cdot 4 + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.2.2

Multiply $x$ by $x$ .

$(4 x^{3} y + 4 x^{2} y \cdot 4 + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.3

Multiply $4$ by $4$ .

$(4 x^{3} y + 16 x^{2} y + 4 x y \cdot 4 + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.4

Multiply $4$ by $4$ .

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 8 y (4 x) + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.5

Rewrite using the commutative property of multiplication.

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 8 \cdot 4 y x + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.6

Multiply $8$ by $4$ .

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 32 y x + 8 y \cdot 4 - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.7

Multiply $4$ by $8$ .

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 32 y x + 32 y - 6 x^{2} - 6 (4 x) - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.8

Multiply $4$ by $- 6$ .

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 32 y x + 32 y - 6 x^{2} - 24 x - 6 \cdot 4) d x + {(x + 2)}^{2} d y = 0$

Step 6.5.9

Multiply $- 6$ by $4$ .

$(4 x^{3} y + 16 x^{2} y + 16 x y + 8 y x^{2} + 32 y x + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} d y = 0$

Step 6.6

Add $16 x^{2} y$ and $8 y x^{2}$ .

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Step 6.6.1

Move $y$ .

$(4 x^{3} y + 16 x^{2} y + 8 x^{2} y + 16 x y + 32 y x + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} d y = 0$

Step 6.6.2

Add $16 x^{2} y$ and $8 x^{2} y$ .

$(4 x^{3} y + 24 x^{2} y + 16 x y + 32 y x + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} d y = 0$

Step 6.7

Add $16 x y$ and $32 y x$ .

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Step 6.7.1

Move $y$ .

$(4 x^{3} y + 24 x^{2} y + 16 x y + 32 x y + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} d y = 0$

Step 6.7.2

Add $16 x y$ and $32 x y$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} d y = 0$

Step 6.8

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + {(x + 2)}^{2} (x^{2} + 4 x + 4) d y = 0$

Step 6.9

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x + 2) (x + 2) (x^{2} + 4 x + 4) d y = 0$

Step 6.10

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

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Step 6.10.1

Apply the distributive property.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x (x + 2) + 2 (x + 2)) (x^{2} + 4 x + 4) d y = 0$

Step 6.10.2

Apply the distributive property.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x \cdot x + x \cdot 2 + 2 (x + 2)) (x^{2} + 4 x + 4) d y = 0$

Step 6.10.3

Apply the distributive property.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) (x^{2} + 4 x + 4) d y = 0$

Step 6.11

Simplify and combine like terms.

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Step 6.11.1

Simplify each term.

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Step 6.11.1.1

Multiply $x$ by $x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) (x^{2} + 4 x + 4) d y = 0$

Step 6.11.1.2

Move $2$ to the left of $x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) (x^{2} + 4 x + 4) d y = 0$

Step 6.11.1.3

Multiply $2$ by $2$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2} + 2 x + 2 x + 4) (x^{2} + 4 x + 4) d y = 0$

Step 6.11.2

Add $2 x$ and $2 x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2} + 4 x + 4) (x^{2} + 4 x + 4) d y = 0$

Step 6.12

Expand $(x^{2} + 4 x + 4) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2} x^{2} + x^{2} (4 x) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13

Simplify each term.

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Step 6.13.1

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

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Step 6.13.1.1

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{2 + 2} + x^{2} (4 x) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.1.2

Add $2$ and $2$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + x^{2} (4 x) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.2

Rewrite using the commutative property of multiplication.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{2} x + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.3

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

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Step 6.13.3.1

Move $x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 (x \cdot x^{2}) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.3.2

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

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Step 6.13.3.2.1

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 (x^{1} x^{2}) + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.3.2.2

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{1 + 2} + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.3.3

Add $1$ and $2$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + x^{2} \cdot 4 + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.4

Move $4$ to the left of $x^{2}$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 \cdot x^{2} + 4 x \cdot x^{2} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.5

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

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Step 6.13.5.1

Move $x^{2}$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 (x^{2} x) + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.5.2

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

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Step 6.13.5.2.1

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 (x^{2} x^{1}) + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.5.2.2

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{2 + 1} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.5.3

Add $2$ and $1$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 4 x (4 x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.6

Rewrite using the commutative property of multiplication.

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 4 \cdot 4 x \cdot x + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.7

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

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Step 6.13.7.1

Move $x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 4 \cdot 4 (x \cdot x) + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.7.2

Multiply $x$ by $x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 4 \cdot 4 x^{2} + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.8

Multiply $4$ by $4$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 16 x^{2} + 4 x \cdot 4 + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.9

Multiply $4$ by $4$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 4 x^{2} + 4 (4 x) + 4 \cdot 4) d y = 0$

Step 6.13.10

Multiply $4$ by $4$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 4 x^{2} + 16 x + 4 \cdot 4) d y = 0$

Step 6.13.11

Multiply $4$ by $4$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 4 x^{3} + 4 x^{2} + 4 x^{3} + 16 x^{2} + 16 x + 4 x^{2} + 16 x + 16) d y = 0$

Step 6.14

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

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 8 x^{3} + 4 x^{2} + 16 x^{2} + 16 x + 4 x^{2} + 16 x + 16) d y = 0$

Step 6.15

Add $4 x^{2}$ and $16 x^{2}$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 4 x^{2} + 16 x + 16) d y = 0$

Step 6.16

Add $20 x^{2}$ and $4 x^{2}$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 8 x^{3} + 24 x^{2} + 16 x + 16 x + 16) d y = 0$

Step 6.17

Add $16 x$ and $16 x$ .

$(4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24) d x + (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) d y = 0$

Step 7

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

$f (x, y) = \int x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16 d y$

Step 8

Integrate $N (x, y) = x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y + C$

Step 9

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

$f (x, y) = (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y + g (x)$

Step 10

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

$\frac{\partial f}{\partial x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y + g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.2

By the Sum Rule, the derivative of $(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y + g (x)$ with respect to $x$ is $\frac{d}{d x} [(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3

Evaluate $\frac{d}{d x} [(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y]$ .

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Step 11.3.1

Since $y$ is constant with respect to $x$ , the derivative of $(x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y$ with respect to $x$ is $y \frac{d}{d x} [x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16]$ .

$y \frac{d}{d x} [x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16] + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.2

By the Sum Rule, the derivative of $x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]$ .

$y (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.3

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

$y (4 x^{3} + \frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.4

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$y (4 x^{3} + 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.5

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

$y (4 x^{3} + 8 (3 x^{2}) + \frac{d}{d x} [24 x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.6

Since $24$ is constant with respect to $x$ , the derivative of $24 x^{2}$ with respect to $x$ is $24 \frac{d}{d x} [x^{2}]$ .

$y (4 x^{3} + 8 (3 x^{2}) + 24 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.7

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

$y (4 x^{3} + 8 (3 x^{2}) + 24 (2 x) + \frac{d}{d x} [32 x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.8

Since $32$ is constant with respect to $x$ , the derivative of $32 x$ with respect to $x$ is $32 \frac{d}{d x} [x]$ .

$y (4 x^{3} + 8 (3 x^{2}) + 24 (2 x) + 32 \frac{d}{d x} [x] + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.9

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

$y (4 x^{3} + 8 (3 x^{2}) + 24 (2 x) + 32 \cdot 1 + \frac{d}{d x} [16]) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.10

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$y (4 x^{3} + 8 (3 x^{2}) + 24 (2 x) + 32 \cdot 1 + 0) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.11

Multiply $3$ by $8$ .

$y (4 x^{3} + 24 x^{2} + 24 (2 x) + 32 \cdot 1 + 0) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.12

Multiply $2$ by $24$ .

$y (4 x^{3} + 24 x^{2} + 48 x + 32 \cdot 1 + 0) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.13

Multiply $32$ by $1$ .

$y (4 x^{3} + 24 x^{2} + 48 x + 32 + 0) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.3.14

Add $4 x^{3} + 24 x^{2} + 48 x + 32$ and $0$ .

$y (4 x^{3} + 24 x^{2} + 48 x + 32) + \frac{d}{d x} [g (x)] = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.4

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

$y (4 x^{3} + 24 x^{2} + 48 x + 32) + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5

Simplify.

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Step 11.5.1

Apply the distributive property.

$y (4 x^{3}) + y (24 x^{2}) + y (48 x) + y \cdot 32 + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5.2

Combine terms.

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Step 11.5.2.1

Move $4$ to the left of $y$ .

$4 \cdot y x^{3} + y (24 x^{2}) + y (48 x) + y \cdot 32 + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5.2.2

Move $24$ to the left of $y$ .

$4 y x^{3} + 24 \cdot y x^{2} + y (48 x) + y \cdot 32 + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5.2.3

Move $48$ to the left of $y$ .

$4 y x^{3} + 24 y x^{2} + 48 \cdot y x + y \cdot 32 + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5.2.4

Move $32$ to the left of $y$ .

$4 y x^{3} + 24 y x^{2} + 48 y x + 32 y + \frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

Step 11.5.3

Reorder terms.

$\frac{d g}{d x} + 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$

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 $4 x^{3} y$ from both sides of the equation.

$\frac{d g}{d x} + 24 x^{2} y + 48 x y + 32 y = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 4 x^{3} y$

Step 12.1.2

Subtract $24 x^{2} y$ from both sides of the equation.

$\frac{d g}{d x} + 48 x y + 32 y = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 4 x^{3} y - 24 x^{2} y$

Step 12.1.3

Subtract $48 x y$ from both sides of the equation.

$\frac{d g}{d x} + 32 y = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 4 x^{3} y - 24 x^{2} y - 48 x y$

Step 12.1.4

Subtract $32 y$ from both sides of the equation.

$\frac{d g}{d x} = 4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 4 x^{3} y - 24 x^{2} y - 48 x y - 32 y$

Step 12.1.5

Combine the opposite terms in $4 x^{3} y + 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 4 x^{3} y - 24 x^{2} y - 48 x y - 32 y$ .

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Step 12.1.5.1

Subtract $4 x^{3} y$ from $4 x^{3} y$ .

$\frac{d g}{d x} = 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 + 0 - 24 x^{2} y - 48 x y - 32 y$

Step 12.1.5.2

Add $24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24$ and $0$ .

$\frac{d g}{d x} = 24 x^{2} y + 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 24 x^{2} y - 48 x y - 32 y$

Step 12.1.5.3

Subtract $24 x^{2} y$ from $24 x^{2} y$ .

$\frac{d g}{d x} = 48 x y + 32 y - 6 x^{2} - 24 x - 24 + 0 - 48 x y - 32 y$

Step 12.1.5.4

Add $48 x y + 32 y - 6 x^{2} - 24 x - 24$ and $0$ .

$\frac{d g}{d x} = 48 x y + 32 y - 6 x^{2} - 24 x - 24 - 48 x y - 32 y$

Step 12.1.5.5

Subtract $48 x y$ from $48 x y$ .

$\frac{d g}{d x} = 32 y - 6 x^{2} - 24 x - 24 + 0 - 32 y$

Step 12.1.5.6

Add $32 y - 6 x^{2} - 24 x - 24$ and $0$ .

$\frac{d g}{d x} = 32 y - 6 x^{2} - 24 x - 24 - 32 y$

Step 12.1.5.7

Subtract $32 y$ from $32 y$ .

$\frac{d g}{d x} = - 6 x^{2} - 24 x - 24 + 0$

Step 12.1.5.8

Add $- 6 x^{2} - 24 x - 24$ and $0$ .

$\frac{d g}{d x} = - 6 x^{2} - 24 x - 24$

Step 13

Find the antiderivative of $- 6 x^{2} - 24 x - 24$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = - 6 x^{2} - 24 x - 24$ .

$\int \frac{d g}{d x} d x = \int - 6 x^{2} - 24 x - 24 d x$

Step 13.2

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

$g (x) = \int - 6 x^{2} - 24 x - 24 d x$

Step 13.3

Split the single integral into multiple integrals.

$g (x) = \int - 6 x^{2} d x + \int - 24 x d x + \int - 24 d x$

Step 13.4

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$g (x) = - 6 \int x^{2} d x + \int - 24 x d x + \int - 24 d x$

Step 13.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$g (x) = - 6 (\frac{1}{3} x^{3} + C) + \int - 24 x d x + \int - 24 d x$

Step 13.6

Since $- 24$ is constant with respect to $x$ , move $- 24$ out of the integral.

$g (x) = - 6 (\frac{1}{3} x^{3} + C) - 24 \int x d x + \int - 24 d x$

Step 13.7

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

$g (x) = - 6 (\frac{1}{3} x^{3} + C) - 24 (\frac{1}{2} x^{2} + C) + \int - 24 d x$

Step 13.8

Apply the constant rule.

$g (x) = - 6 (\frac{1}{3} x^{3} + C) - 24 (\frac{1}{2} x^{2} + C) - 24 x + C$

Step 13.9

Simplify.

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Step 13.9.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$g (x) = - 6 (\frac{x^{3}}{3} + C) - 24 (\frac{1}{2} x^{2} + C) - 24 x + C$

Step 13.9.2

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

$g (x) = - 6 (\frac{x^{3}}{3} + C) - 24 (\frac{x^{2}}{2} + C) - 24 x + C$

Step 13.10

Simplify.

$g (x) = - 2 x^{3} - 12 x^{2} - 24 x + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y + g (x)$ .

$f (x, y) = (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16) y - 2 x^{3} - 12 x^{2} - 24 x + C$

Step 15

Apply the distributive property.

$f (x, y) = x^{4} y + 8 x^{3} y + 24 x^{2} y + 32 x y + 16 y - 2 x^{3} - 12 x^{2} - 24 x + C$