Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = y^{2} (0 + \frac{d}{d y} [y] + \frac{d}{d y} [1]) + (x + y + 1) \frac{d}{d y} [y^{2}]$

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

$\frac{\partial M}{\partial y} = y^{2} (1 + 0) + (x + y + 1) \frac{d}{d y} [y^{2}]$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $1$ and $0$ .

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

Step 1.3.6.2

Multiply $y^{2}$ by $1$ .

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

Step 1.3.7

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

$\frac{\partial M}{\partial y} = y^{2} + (x + y + 1) (2 y)$

Step 1.3.8

Move $2$ to the left of $x + y + 1$ .

$\frac{\partial M}{\partial y} = y^{2} + 2 \cdot (x + y + 1) y$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$\frac{\partial M}{\partial y} = y^{2} + (2 x + 2 y + 2 \cdot 1) y$

Step 1.4.2

Apply the distributive property.

$\frac{\partial M}{\partial y} = y^{2} + 2 x y + 2 y \cdot y + 2 \cdot 1 y$

Step 1.4.3

Combine terms.

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Step 1.4.3.1

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

$\frac{\partial M}{\partial y} = y^{2} + 2 x y + 2 (y^{1} y) + 2 \cdot 1 y$

Step 1.4.3.2

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

$\frac{\partial M}{\partial y} = y^{2} + 2 x y + 2 (y^{1} y^{1}) + 2 \cdot 1 y$

Step 1.4.3.3

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

$\frac{\partial M}{\partial y} = y^{2} + 2 x y + 2 y^{1 + 1} + 2 \cdot 1 y$

Step 1.4.3.4

Add $1$ and $1$ .

$\frac{\partial M}{\partial y} = y^{2} + 2 x y + 2 y^{2} + 2 \cdot 1 y$

Step 1.4.3.5

Multiply $2$ by $1$ .

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

Step 1.4.3.6

Add $y^{2}$ and $2 y^{2}$ .

$\frac{\partial M}{\partial y} = 3 y^{2} + 2 x y + 2 y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

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 $g (x) = x + 3 y + 2$ .

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

Step 2.4

Differentiate.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

$\frac{\partial N}{\partial x} = y (x (1 + 0 + \frac{d}{d x} [2]) + (x + 3 y + 2) \frac{d}{d x} [x])$

Step 2.4.4

Add $1$ and $0$ .

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

Step 2.4.5

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

$\frac{\partial N}{\partial x} = y (x (1 + 0) + (x + 3 y + 2) \frac{d}{d x} [x])$

Step 2.4.6

Simplify the expression.

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Step 2.4.6.1

Add $1$ and $0$ .

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

Step 2.4.6.2

Multiply $x$ by $1$ .

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

Step 2.4.7

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

$\frac{\partial N}{\partial x} = y (x + (x + 3 y + 2) \cdot 1)$

Step 2.4.8

Simplify by adding terms.

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Step 2.4.8.1

Multiply $x + 3 y + 2$ by $1$ .

$\frac{\partial N}{\partial x} = y (x + x + 3 y + 2)$

Step 2.4.8.2

Add $x$ and $x$ .

$\frac{\partial N}{\partial x} = y (2 x + 3 y + 2)$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$\frac{\partial N}{\partial x} = y (2 x) + y (3 y) + y \cdot 2$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

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

$\frac{\partial N}{\partial x} = 2 \cdot y x + y (3 y) + y \cdot 2$

Step 2.5.2.2

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

$\frac{\partial N}{\partial x} = 2 y x + 3 (y^{1} y) + y \cdot 2$

Step 2.5.2.3

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

$\frac{\partial N}{\partial x} = 2 y x + 3 (y^{1} y^{1}) + y \cdot 2$

Step 2.5.2.4

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

$\frac{\partial N}{\partial x} = 2 y x + 3 y^{1 + 1} + y \cdot 2$

Step 2.5.2.5

Add $1$ and $1$ .

$\frac{\partial N}{\partial x} = 2 y x + 3 y^{2} + y \cdot 2$

Step 2.5.2.6

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

$\frac{\partial N}{\partial x} = 2 y x + 3 y^{2} + 2 y$

Step 2.5.3

Reorder terms.

$\frac{\partial N}{\partial x} = 3 y^{2} + 2 x y + 2 y$

Step 3

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

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Step 3.1

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

$3 y^{2} + 2 x y + 2 y = 3 y^{2} + 2 x y + 2 y$

Step 3.2

Since the two sides have been shown to be equivalent, the equation is an identity.

$3 y^{2} + 2 x y + 2 y = 3 y^{2} + 2 x y + 2 y$ is an identity.

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

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

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

Step 5.4

Apply the constant rule.

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

Step 5.5

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

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

Step 5.6

Apply the constant rule.

$f (x, y) = y^{2} (\frac{x^{2}}{2} + C + y x + C + x + C)$

Step 5.7

Simplify.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial y} = x y (x + 3 y + 2)$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

Differentiate using the Sum Rule.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Since $\frac{x^{2}}{2}$ is constant with respect to $y$ , the derivative of $\frac{x^{2}}{2}$ with respect to $y$ is $0$ .

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

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

Step 8.3.7

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

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

Step 8.3.8

Multiply $x$ by $1$ .

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

Step 8.3.9

Add $0$ and $x$ .

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

Step 8.3.10

Add $x$ and $0$ .

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

Step 8.3.11

Move $2$ to the left of $\frac{x^{2}}{2} + y x + x$ .

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

Step 8.4

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

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

Step 8.5

Simplify.

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Step 8.5.1

Apply the distributive property.

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

Step 8.5.2

Apply the distributive property.

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

Step 8.5.3

Combine terms.

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Step 8.5.3.1

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

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

Step 8.5.3.2

Cancel the common factor of $2$ .

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Step 8.5.3.2.1

Cancel the common factor.

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

Step 8.5.3.2.2

Divide $x^{2}$ by $1$ .

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

Step 8.5.3.3

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

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

Step 8.5.3.4

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

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

Step 8.5.3.5

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

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

Step 8.5.3.6

Add $1$ and $1$ .

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

Step 8.5.3.7

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

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Step 8.5.3.7.1

Reorder $y^{2}$ and $x$ .

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

Step 8.5.3.7.2

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

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

Step 8.5.4

Reorder terms.

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

Simplify $x y (x + 3 y + 2)$ .

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Step 9.1.1.1

Rewrite.

$\frac{d g}{d y} + 3 y^{2} x + x^{2} y + 2 x y = 0 + 0 + x y (x + 3 y + 2)$

Step 9.1.1.2

Simplify by adding zeros.

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

Step 9.1.1.3

Apply the distributive property.

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

Step 9.1.1.4

Simplify.

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Step 9.1.1.4.1

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

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Step 9.1.1.4.1.1

Move $x$ .

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

Step 9.1.1.4.1.2

Multiply $x$ by $x$ .

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

Step 9.1.1.4.2

Rewrite using the commutative property of multiplication.

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

Step 9.1.1.4.3

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

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

Step 9.1.1.5

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

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Step 9.1.1.5.1

Move $y$ .

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

Step 9.1.1.5.2

Multiply $y$ by $y$ .

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

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

Step 9.1.2

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

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Step 9.1.2.1

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

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

Step 9.1.2.2

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

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

Step 9.1.2.3

Subtract $2 x y$ from both sides of the equation.

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

Step 9.1.2.4

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

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Step 9.1.2.4.1

Reorder the factors in the terms $3 x y^{2}$ and $- 3 y^{2} x$ .

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

Step 9.1.2.4.2

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

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

Step 9.1.2.4.3

Add $x^{2} y + 2 x y$ and $0$ .

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

Step 9.1.2.4.4

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

$\frac{d g}{d y} = 2 x y + 0 - 2 x y$

Step 9.1.2.4.5

Add $2 x y$ and $0$ .

$\frac{d g}{d y} = 2 x y - 2 x y$

Step 9.1.2.4.6

Subtract $2 x y$ from $2 x y$ .

$\frac{d g}{d y} = 0$

Step 10

Find the antiderivative of $0$ to find $g (y)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

$\int \frac{d g}{d y} d y = \int 0 d y$

Step 10.2

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

$g (y) = \int 0 d y$

Step 10.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 10.4

Add $0$ and $C$ .

$g (y) = C$

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

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

$f (x, y) = y^{2} (\frac{x^{2}}{2} + y x + x) + C$

Step 12.2

Apply the distributive property.

$f (x, y) = y^{2} \frac{x^{2}}{2} + y^{2} (y x) + y^{2} x + C$

Step 12.3

Simplify.

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Step 12.3.1

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

$f (x, y) = \frac{y^{2} x^{2}}{2} + y^{2} (y x) + y^{2} x + C$

Step 12.3.2

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

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Step 12.3.2.1

Move $y$ .

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

Step 12.3.2.2

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

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Step 12.3.2.2.1

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

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

Step 12.3.2.2.2

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

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

Step 12.3.2.3

Add $1$ and $2$ .

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