Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 1$ by $1$ .

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

Step 1.4

Reorder terms.

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 1$ by $1$ .

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

Step 3

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

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Step 3.1

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

$- x + 2 y = 2 x - y$

Step 3.2

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

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

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

Step 4.3.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.2.3

Multiply $- - y$ .

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Step 4.3.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2.3.2

Multiply $y$ by $1$ .

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

Step 4.3.2.4

Subtract $2 x$ from $- x$ .

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

Step 4.3.2.5

Add $2 y$ and $y$ .

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

Step 4.3.2.6

Factor $3$ out of $- 3 x + 3 y$ .

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Step 4.3.2.6.1

Factor $3$ out of $- 3 x$ .

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

Step 4.3.2.6.2

Factor $3$ out of $3 y$ .

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

Step 4.3.2.6.3

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

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

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

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

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Step 4.3.3.1.1

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

$\frac{3 (- x + y)}{x \cdot x - x y}$

Step 4.3.3.1.2

Factor $x$ out of $- x y$ .

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

Step 4.3.3.1.3

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

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

Step 4.3.3.2

Rewrite $- 1 y$ as $- y$ .

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

Step 4.3.4

Cancel the common factor of $- x + y$ and $x - y$ .

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Step 4.3.4.1

Factor $- 1$ out of $- x$ .

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

Step 4.3.4.2

Factor $- 1$ out of $y$ .

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

Step 4.3.4.3

Factor $- 1$ out of $- (x) - 1 (- y)$ .

$\frac{3 (- (x - y))}{x (x - y)}$

Step 4.3.4.4

Rewrite $- (x - y)$ as $- 1 (x - y)$ .

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

Step 4.3.4.5

Cancel the common factor.

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

Step 4.3.4.6

Rewrite the expression.

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

Step 4.3.5

Multiply $3$ by $- 1$ .

$\frac{- 3}{x}$

Step 4.3.6

Move the negative in front of the fraction.

$- \frac{3}{x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{3}{x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{3}{x} d x}$ .

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Step 5.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$μ (x, y) = e^{- \int \frac{3}{x} d x}$

Step 5.2

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

$μ (x, y) = e^{- (3 \int \frac{1}{x} d x)}$

Step 5.3

Multiply $3$ by $- 1$ .

$μ (x, y) = e^{- 3 \int \frac{1}{x} d x}$

Step 5.4

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

$μ (x, y) = e^{- 3 (\ln (| x |) + C)}$

Step 5.5

Simplify.

$μ (x, y) = e^{- 3 \ln (| x |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $- 3 \ln (| x |)$ by moving $- 3$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{- 3})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 3} + C$

Step 5.6.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{{| x |}^{3}} + C$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Factor $y$ out of $- x y$ .

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

Step 6.3.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

Simplify the numerator.

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Step 6.6.1

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

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Step 6.6.1.1

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

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

Step 6.6.1.2

Factor $x$ out of $- x y$ .

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

Step 6.6.1.3

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

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

Step 6.6.2

Rewrite $- 1 y$ as $- y$ .

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

Step 6.7

Cancel the common factors.

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Step 6.7.1

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

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

Step 6.7.2

Cancel the common factor.

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

Step 6.7.3

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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Step 8.1

Since $\frac{1}{x^{2}}$ is constant with respect to $y$ , move $\frac{1}{x^{2}}$ out of the integral.

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

Step 8.2

Split the single integral into multiple integrals.

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

Step 8.3

Apply the constant rule.

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

Step 8.4

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 8.5

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

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

Step 8.6

Simplify.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

Differentiate using the Sum Rule.

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

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

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Step 11.3.1

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 $f (x) = \frac{1}{x^{2}}$ and $g (x) = x y - \frac{y^{2}}{2}$ .

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

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

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

Step 11.3.5

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

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

Step 11.3.6

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 11.3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 11.3.7.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

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

Step 11.3.7.2

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

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

Step 11.3.7.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 11.3.8

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

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

Step 11.3.9

Multiply $y$ by $1$ .

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

Step 11.3.10

Add $y$ and $0$ .

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

Step 11.3.11

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

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

Step 11.3.12

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 11.3.12.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 11.3.12.2

Multiply $2$ by $- 2$ .

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

Step 11.3.13

Multiply $2$ by $- 1$ .

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

Step 11.3.14

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

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

Step 11.3.15

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

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

Step 11.3.16

Subtract $4$ from $1$ .

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

Step 11.3.17

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

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

Step 11.3.18

Combine the numerators over the common denominator.

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

Step 11.3.19

Multiply $x^{- 3}$ by $x^{2}$ by adding the exponents.

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Step 11.3.19.1

Move $x^{2}$ .

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

Step 11.3.19.2

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

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

Step 11.3.19.3

Subtract $3$ from $2$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.5.2

Apply the distributive property.

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

Step 11.5.3

Combine terms.

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Step 11.5.3.1

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

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

Step 11.5.3.2

Move the negative in front of the fraction.

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

Step 11.5.3.3

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

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

Step 11.5.3.4

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

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

Step 11.5.3.5

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

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

Step 11.5.3.6

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

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

Step 11.5.3.7

Cancel the common factor of $x$ .

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Step 11.5.3.7.1

Cancel the common factor.

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

Step 11.5.3.7.2

Divide $2 y$ by $1$ .

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

Step 11.5.3.8

Multiply $2$ by $- 1$ .

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

Step 11.5.3.9

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

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

Step 11.5.3.10

Move the negative in front of the fraction.

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

Step 11.5.3.11

Multiply $- 1$ by $- 1$ .

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

Step 11.5.3.12

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

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

Step 11.5.3.13

Multiply $\frac{y^{2}}{2}$ by $\frac{2}{x}$ .

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

Step 11.5.3.14

Move $2$ to the left of $y^{2}$ .

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

Step 11.5.3.15

Cancel the common factor of $2$ .

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Step 11.5.3.15.1

Cancel the common factor.

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

Step 11.5.3.15.2

Rewrite the expression.

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

Step 11.5.3.16

Subtract $2 y$ from $y$ .

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

Step 11.5.3.17

Multiply $\frac{- y + \frac{y^{2}}{x}}{x^{2}}$ by $\frac{x}{x}$ .

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

Step 11.5.3.18

Combine.

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

Step 11.5.3.19

Apply the distributive property.

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

Step 11.5.3.20

Cancel the common factor of $x$ .

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Step 11.5.3.20.1

Cancel the common factor.

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

Step 11.5.3.20.2

Rewrite the expression.

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

Step 11.5.3.21

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

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Step 11.5.3.21.1

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

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Step 11.5.3.21.1.1

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

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

Step 11.5.3.21.1.2

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

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

Step 11.5.3.21.2

Add $1$ and $2$ .

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

Step 11.5.3.22

To write $\frac{d g}{d x}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

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

Step 11.5.3.23

Combine the numerators over the common denominator.

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

Step 11.5.4

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

Step 12.1.2

Simplify $y (y - x)$ .

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Step 12.1.2.1

Rewrite.

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

Step 12.1.2.2

Simplify by adding zeros.

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

Step 12.1.2.3

Apply the distributive property.

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

Step 12.1.2.4

Simplify the expression.

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Step 12.1.2.4.1

Multiply $y$ by $y$ .

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

Step 12.1.2.4.2

Rewrite using the commutative property of multiplication.

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

Step 12.1.3

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

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Step 12.1.3.1

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

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

Step 12.1.3.2

Add $x y$ to both sides of the equation.

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

Step 12.1.3.3

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

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Step 12.1.3.3.1

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

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

Step 12.1.3.3.2

Add $- y x$ and $0$ .

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

Step 12.1.3.3.3

Reorder the factors in the terms $- y x$ and $x y$ .

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

Step 12.1.3.3.4

Add $- x y$ and $x y$ .

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

Step 12.1.4

Divide each term in $x^{3} \frac{d g}{d x} = 0$ by $x^{3}$ and simplify.

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Step 12.1.4.1

Divide each term in $x^{3} \frac{d g}{d x} = 0$ by $x^{3}$ .

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

Step 12.1.4.2

Simplify the left side.

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Step 12.1.4.2.1

Cancel the common factor of $x^{3}$ .

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Step 12.1.4.2.1.1

Cancel the common factor.

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

Step 12.1.4.2.1.2

Divide $\frac{d g}{d x}$ by $1$ .

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

Step 12.1.4.3

Simplify the right side.

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Step 12.1.4.3.1

Divide $0$ by $x^{3}$ .

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

$g (x) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (x) = C$

Step 14

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

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

Simplify the numerator.

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Step 15.3.1

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

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Step 15.3.1.1

Factor $y$ out of $x y$ .

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

Step 15.3.1.2

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

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

Step 15.3.1.3

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

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

Step 15.3.2

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

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

Step 15.3.3

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

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

Step 15.3.4

Combine the numerators over the common denominator.

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

Step 15.3.5

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

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

Step 15.4

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

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

Step 15.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.6

Combine.

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

Step 15.7

Multiply $y$ by $1$ .

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