Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $1$ .

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

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

Step 1.4.2

Add $2 x$ and $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = 0 + \frac{d}{d x} [x^{2}]$

Step 2.4

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

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

Step 2.5

Add $0$ and $2 x$ .

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

Step 3

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

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Step 3.1

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

$2 x = 2 x$

Step 3.2

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

$2 x = 2 x$ is an identity.

Step 4

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

$f (x, y) = \int 2 x y - x d x$

Step 5

Integrate $M (x, y) = 2 x y - x$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int 2 x y d x + \int - x d x$

Step 5.2

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

$f (x, y) = 2 y \int x d x + \int - x d x$

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Simplify.

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

Step 5.7

Simplify.

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Step 5.7.1

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

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

Step 5.7.2

Cancel the common factor of $2$ .

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Step 5.7.2.1

Cancel the common factor.

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

Step 5.7.2.2

Rewrite the expression.

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

Step 5.7.3

Multiply $y$ by $1$ .

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

Step 5.7.4

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

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

Step 5.7.5

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

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

Step 5.7.6

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

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

Step 5.7.7

Combine the numerators over the common denominator.

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

Step 5.7.8

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

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

Step 5.7.9

Remove parentheses.

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

Step 5.8

Reorder terms.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

Multiply $2$ by $1$ .

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

Step 8.3.7

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

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

Step 8.3.8

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

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

Step 8.3.9

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

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

Step 8.3.10

Cancel the common factor of $2$ .

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Step 8.3.10.1

Cancel the common factor.

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

Step 8.3.10.2

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

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

Step 8.4

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

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

Step 8.5

Reorder terms.

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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

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Step 9.1.2.1

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

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

Step 9.1.2.2

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

$\frac{d g}{d y} = y^{2}$

Step 10

Find the antiderivative of $y^{2}$ to find $g (y)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d y} = y^{2}$ .

$\int \frac{d g}{d y} d y = \int y^{2} d y$

Step 10.2

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

$g (y) = \int y^{2} d y$

Step 10.3

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

$g (y) = \frac{1}{3} y^{3} + C$

Step 11

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

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

Step 12

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

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Step 12.1

Simplify each term.

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Step 12.1.1

Apply the distributive property.

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

Step 12.1.2

Cancel the common factor of $2$ .

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Step 12.1.2.1

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

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

Step 12.1.2.2

Cancel the common factor.

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

Step 12.1.2.3

Rewrite the expression.

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

Step 12.1.3

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

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

Step 12.1.4

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

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

Step 12.1.5

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

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

Step 12.1.6

Combine the numerators over the common denominator.

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

Step 12.1.7

Simplify the numerator.

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Step 12.1.7.1

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

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Step 12.1.7.1.1

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

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

Step 12.1.7.1.2

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

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

Step 12.1.7.1.3

Factor $x^{2}$ out of $x^{2} (y \cdot 2) + x^{2} \cdot - 1$ .

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

Step 12.1.7.2

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

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

Step 12.1.8

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

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.4.1

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

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

Step 12.4.2

Multiply $2$ by $3$ .

$f (x, y) = \frac{x^{2} (2 y - 1) \cdot 3}{6} + \frac{y^{3}}{3} \cdot \frac{2}{2} + C$

Step 12.4.3

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

$f (x, y) = \frac{x^{2} (2 y - 1) \cdot 3}{6} + \frac{y^{3} \cdot 2}{3 \cdot 2} + C$

Step 12.4.4

Multiply $3$ by $2$ .

$f (x, y) = \frac{x^{2} (2 y - 1) \cdot 3}{6} + \frac{y^{3} \cdot 2}{6} + C$

Step 12.5

Combine the numerators over the common denominator.

$f (x, y) = \frac{x^{2} (2 y - 1) \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6

Simplify the numerator.

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Step 12.6.1

Apply the distributive property.

$f (x, y) = \frac{(x^{2} (2 y) + x^{2} \cdot - 1) \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.2

Rewrite using the commutative property of multiplication.

$f (x, y) = \frac{(2 x^{2} y + x^{2} \cdot - 1) \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.3

Move $- 1$ to the left of $x^{2}$ .

$f (x, y) = \frac{(2 x^{2} y - 1 \cdot x^{2}) \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f (x, y) = \frac{(2 x^{2} y - x^{2}) \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.5

Apply the distributive property.

$f (x, y) = \frac{2 x^{2} y \cdot 3 - x^{2} \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.6

Multiply $3$ by $2$ .

$f (x, y) = \frac{6 x^{2} y - x^{2} \cdot 3 + y^{3} \cdot 2}{6} + C$

Step 12.6.7

Multiply $3$ by $- 1$ .

$f (x, y) = \frac{6 x^{2} y - 3 x^{2} + y^{3} \cdot 2}{6} + C$

Step 12.6.8

Move $2$ to the left of $y^{3}$ .

$f (x, y) = \frac{6 x^{2} y - 3 x^{2} + 2 y^{3}}{6} + C$