Calculus Examples

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

Step 1

Rewrite the differential equation to fit the Exact differential equation technique.

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Step 1.1

Rewrite.

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

Step 2

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

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Step 2.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x y^{2} - y]$

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

Multiply $- 1$ by $1$ .

$\frac{\partial M}{\partial y} = 2 x y - 1$

Step 3

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

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Step 3.1

Differentiate $N$ with respect to $x$ .

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

Step 3.2

Differentiate using the Constant Multiple Rule.

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Step 3.2.1

Factor $x$ out of $3 x^{2} y - 4 x$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Factor $x$ out of $- 4 x$ .

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

Step 3.2.1.3

Factor $x$ out of $x (3 x y) + x \cdot - 4$ .

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

Step 3.2.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x (3 x y - 4)}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x (3 x y - 4)]$ .

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

Step 3.3

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 $g (x) = 3 x y - 4$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Multiply $3$ by $1$ .

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

Step 3.4.5

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

$\frac{\partial N}{\partial x} = \frac{1}{2} (x (3 y + 0) + (3 x y - 4) \frac{d}{d x} [x])$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Add $3 y$ and $0$ .

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

Step 3.4.6.2

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

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

Step 3.4.7

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

$\frac{\partial N}{\partial x} = \frac{1}{2} (3 x y + (3 x y - 4) \cdot 1)$

Step 3.4.8

Simplify by adding terms.

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Step 3.4.8.1

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

$\frac{\partial N}{\partial x} = \frac{1}{2} (3 x y + 3 x y - 4)$

Step 3.4.8.2

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

$\frac{\partial N}{\partial x} = \frac{1}{2} (6 x y - 4)$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$\frac{\partial N}{\partial x} = \frac{1}{2} (6 x y) + \frac{1}{2} \cdot - 4$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $6$ and $\frac{1}{2}$ .

$\frac{\partial N}{\partial x} = \frac{6}{2} (x y) + \frac{1}{2} \cdot - 4$

Step 3.5.2.2

Combine $x$ and $\frac{6}{2}$ .

$\frac{\partial N}{\partial x} = \frac{x \cdot 6}{2} y + \frac{1}{2} \cdot - 4$

Step 3.5.2.3

Combine $\frac{x \cdot 6}{2}$ and $y$ .

$\frac{\partial N}{\partial x} = \frac{x \cdot 6 y}{2} + \frac{1}{2} \cdot - 4$

Step 3.5.2.4

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

$\frac{\partial N}{\partial x} = \frac{6 \cdot x y}{2} + \frac{1}{2} \cdot - 4$

Step 3.5.2.5

Cancel the common factor of $6$ and $2$ .

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Step 3.5.2.5.1

Factor $2$ out of $6 x y$ .

$\frac{\partial N}{\partial x} = \frac{2 (3 x y)}{2} + \frac{1}{2} \cdot - 4$

Step 3.5.2.5.2

Cancel the common factors.

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Step 3.5.2.5.2.1

Factor $2$ out of $2$ .

$\frac{\partial N}{\partial x} = \frac{2 (3 x y)}{2 (1)} + \frac{1}{2} \cdot - 4$

Step 3.5.2.5.2.2

Cancel the common factor.

$\frac{\partial N}{\partial x} = \frac{2 (3 x y)}{2 \cdot 1} + \frac{1}{2} \cdot - 4$

Step 3.5.2.5.2.3

Rewrite the expression.

$\frac{\partial N}{\partial x} = \frac{3 x y}{1} + \frac{1}{2} \cdot - 4$

Step 3.5.2.5.2.4

Divide $3 x y$ by $1$ .

$\frac{\partial N}{\partial x} = 3 x y + \frac{1}{2} \cdot - 4$

Step 3.5.2.6

Combine $\frac{1}{2}$ and $- 4$ .

$\frac{\partial N}{\partial x} = 3 x y + \frac{- 4}{2}$

Step 3.5.2.7

Cancel the common factor of $- 4$ and $2$ .

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Step 3.5.2.7.1

Factor $2$ out of $- 4$ .

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

Step 3.5.2.7.2

Cancel the common factors.

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Step 3.5.2.7.2.1

Factor $2$ out of $2$ .

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

Step 3.5.2.7.2.2

Cancel the common factor.

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

Step 3.5.2.7.2.3

Rewrite the expression.

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

Step 3.5.2.7.2.4

Divide $- 2$ by $1$ .

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

Step 4

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

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Step 4.1

Substitute $2 x y - 1$ for $\frac{\partial M}{\partial y}$ and $3 x y - 2$ for $\frac{\partial N}{\partial x}$ .

$2 x y - 1 = 3 x y - 2$

Step 4.2

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

$2 x y - 1 = 3 x y - 2$ is not an identity.

Step 5

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

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Step 5.1

Substitute $3 x y - 2$ for $\frac{\partial N}{\partial x}$ .

$\frac{3 x y - 2 - \frac{\partial M}{\partial y}}{M}$

Step 5.2

Substitute $2 x y - 1$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x y - 2 - (2 x y - 1)}{M}$

Step 5.3

Substitute $x y^{2} - y$ for $M$ .

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Step 5.3.1

Substitute $x y^{2} - y$ for $M$ .

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

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Apply the distributive property.

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

Step 5.3.2.2

Multiply $2$ by $- 1$ .

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

Step 5.3.2.3

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2.4

Subtract $2 x y$ from $3 x y$ .

$\frac{1 x y - 2 + 1}{x y^{2} - y}$

Step 5.3.2.5

Add $- 2$ and $1$ .

$\frac{1 x y - 1}{x y^{2} - y}$

Step 5.3.2.6

Multiply $x$ by $1$ .

$\frac{x y - 1}{x y^{2} - y}$

Step 5.3.3

Factor $y$ out of $x y^{2} - y$ .

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Step 5.3.3.1

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

$\frac{x y - 1}{y (x y) - y}$

Step 5.3.3.2

Factor $y$ out of $- y$ .

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

Step 5.3.3.3

Factor $y$ out of $y (x y) + y \cdot - 1$ .

$\frac{x y - 1}{y (x y - 1)}$

Step 5.3.4

Substitute $x y^{2} - y$ for $M$ .

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Step 5.3.4.1

Cancel the common factor.

$\frac{x y - 1}{y (x y - 1)}$

Step 5.3.4.2

Rewrite the expression.

$\frac{1}{y}$

Step 5.4

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

$μ (x, y) = e^{\int \frac{1}{y} d y}$

Step 6

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

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Step 6.1

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

$μ (x, y) = e^{\ln (| y |) + C}$

Step 6.2

Simplify the answer.

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Step 6.2.1

Simplify.

$μ (x, y) = e^{\ln (| y |)} + C$

Step 6.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | y | + C$

Step 7

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

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

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

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Step 7.3.1

Move $y$ .

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

Step 7.3.2

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

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Step 7.3.2.1

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

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

Step 7.3.2.2

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

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

Step 7.3.3

Add $1$ and $2$ .

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

Step 7.4

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

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Step 7.4.1

Move $y$ .

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

Step 7.4.2

Multiply $y$ by $y$ .

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

Step 7.5

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

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

Step 7.6

Factor $x$ out of $3 x^{2} y - 4 x$ .

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Step 7.6.1

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

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

Step 7.6.2

Factor $x$ out of $- 4 x$ .

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

Step 7.6.3

Factor $x$ out of $x (3 x y) + x \cdot - 4$ .

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

Step 7.7

Combine $\frac{x (3 x y - 4)}{2}$ and $y$ .

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

Step 7.8

Reorder factors in $\frac{x (3 x y - 4) y}{2}$ .

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

Step 8

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

$f (x, y) = \int \frac{x y (3 x y - 4)}{2} d y$

Step 9

Integrate $N (x, y) = \frac{x y (3 x y - 4)}{2}$ to find $f (x, y)$ .

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Step 9.1

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

$f (x, y) = \frac{x}{2} \int y (3 x y - 4) d y$

Step 9.2

Expand $y (3 x y - 4)$ .

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Step 9.2.1

Apply the distributive property.

$f (x, y) = \frac{x}{2} \int y (3 x y) + y \cdot - 4 d y$

Step 9.2.2

Move parentheses.

$f (x, y) = \frac{x}{2} \int y (3 x) y + y \cdot - 4 d y$

Step 9.2.3

Remove parentheses.

$f (x, y) = \frac{x}{2} \int y \cdot 3 x y + y \cdot - 4 d y$

Step 9.2.4

Reorder $y$ and $3$ .

$f (x, y) = \frac{x}{2} \int 3 \cdot y x y + y \cdot - 4 d y$

Step 9.2.5

Reorder $y$ and $- 4$ .

$f (x, y) = \frac{x}{2} \int 3 y x y - 4 y d y$

Step 9.3

Simplify.

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Step 9.3.1

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

$f (x, y) = \frac{x}{2} \int 3 x (y^{1} y) - 4 y d y$

Step 9.3.2

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

$f (x, y) = \frac{x}{2} \int 3 x (y^{1} y^{1}) - 4 y d y$

Step 9.3.3

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

$f (x, y) = \frac{x}{2} \int 3 x y^{1 + 1} - 4 y d y$

Step 9.3.4

Add $1$ and $1$ .

$f (x, y) = \frac{x}{2} \int 3 x y^{2} - 4 y d y$

Step 9.4

Split the single integral into multiple integrals.

$f (x, y) = \frac{x}{2} (\int 3 x y^{2} d y + \int - 4 y d y)$

Step 9.5

Since $3 x$ is constant with respect to $y$ , move $3 x$ out of the integral.

$f (x, y) = \frac{x}{2} (3 x \int y^{2} d y + \int - 4 y d y)$

Step 9.6

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

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

Step 9.7

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

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

Step 9.8

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

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

Step 9.9

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

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

Step 9.10

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

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

Step 9.11

Simplify.

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

Step 9.12

Reorder terms.

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

Step 10

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

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

Step 11

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

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

Step 12

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

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Step 12.1

Differentiate $f$ with respect to $x$ .

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

Step 12.2

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

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

Step 12.3.4

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

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

Step 12.3.5

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

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

Step 12.3.6

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

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

Step 12.3.7

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

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

Step 12.3.8

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

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

Step 12.3.9

Multiply $y^{3}$ by $1$ .

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

Step 12.3.10

Add $y^{3}$ and $0$ .

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

Step 12.3.11

Combine $\frac{x}{2}$ and $y^{3}$ .

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

Step 12.3.12

Multiply $\frac{1}{2}$ by $1$ .

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

Step 12.3.13

To write $(x y^{3} - 2 y^{2}) \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.3.14

Combine $(x y^{3} - 2 y^{2}) \frac{1}{2}$ and $\frac{2}{2}$ .

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

Step 12.3.15

Combine the numerators over the common denominator.

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

Step 12.3.16

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

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

Step 12.3.17

Cancel the common factor of $2$ .

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Step 12.3.17.1

Cancel the common factor.

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

Step 12.3.17.2

Rewrite the expression.

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

Step 12.3.18

Multiply $x y^{3} - 2 y^{2}$ by $1$ .

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

Step 12.3.19

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

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

Step 12.4

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

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

Step 12.5

Simplify.

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Step 12.5.1

Combine terms.

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Step 12.5.1.1

To write $\frac{d g}{d x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 12.5.1.2

Combine $\frac{d g}{d x}$ and $\frac{2}{2}$ .

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

Step 12.5.1.3

Combine the numerators over the common denominator.

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

Step 12.5.1.4

Move $2$ to the left of $\frac{d g}{d x}$ .

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

Step 12.5.1.5

Cancel the common factor of $2 x y^{3} - 2 y^{2} + 2 \frac{d g}{d x}$ and $2$ .

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Step 12.5.1.5.1

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

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

Step 12.5.1.5.2

Factor $2$ out of $- 2 y^{2}$ .

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

Step 12.5.1.5.3

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

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

Step 12.5.1.5.4

Factor $2$ out of $2 \frac{d g}{d x}$ .

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

Step 12.5.1.5.5

Factor $2$ out of $2 (x y^{3} - y^{2}) + 2 (\frac{d g}{d x})$ .

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

Step 12.5.1.5.6

Cancel the common factors.

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Step 12.5.1.5.6.1

Factor $2$ out of $2$ .

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

Step 12.5.1.5.6.2

Cancel the common factor.

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

Step 12.5.1.5.6.3

Rewrite the expression.

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

Step 12.5.1.5.6.4

Divide $x y^{3} - y^{2} + \frac{d g}{d x}$ by $1$ .

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

Step 12.5.2

Reorder terms.

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

Step 13

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

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Step 13.1

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

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Step 13.1.1

Add $y^{2}$ to both sides of the equation.

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

Step 13.1.2

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

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

Step 13.1.3

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

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Step 13.1.3.1

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

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

Step 13.1.3.2

Add $x y^{3}$ and $0$ .

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

Step 13.1.3.3

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

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

Step 13.1.3.4

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

$g (x) = 0 + C$

Step 14.4

Add $0$ and $C$ .

$g (x) = C$

Step 15

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

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

Step 16

Simplify each term.

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Step 16.1

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

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

Step 16.2

Apply the distributive property.

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

Step 16.3

Multiply $\frac{x}{2} (x y^{3})$ .

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Step 16.3.1

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

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

Step 16.3.2

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

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

Step 16.3.3

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

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

Step 16.3.4

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

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

Step 16.3.5

Add $1$ and $1$ .

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

Step 16.3.6

Combine $\frac{x^{2}}{2}$ and $y^{3}$ .

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

Step 16.4

Rewrite using the commutative property of multiplication.

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

Step 16.5

Cancel the common factor of $2$ .

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Step 16.5.1

Factor $2$ out of $- 2$ .

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

Step 16.5.2

Cancel the common factor.

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

Step 16.5.3

Rewrite the expression.

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