Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Differentiate.

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Step 1.4.1

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

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

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 2 x y + 1 + 0$

Step 1.4.3

Add $2 x y + 1$ and $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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 $f (x) = x$ and $g (x) = x y + 1$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $y$ by $1$ .

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

Step 2.3.5

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

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

Step 2.3.6

Add $y$ and $0$ .

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

Step 2.3.7

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

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

Step 2.3.8

Simplify by adding terms.

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Step 2.3.8.1

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

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

Step 2.3.8.2

Add $x y$ and $x y$ .

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

Step 3

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

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Step 3.1

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

$2 x y + 1 = 2 x y + 1$

Step 3.2

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

$2 x y + 1 = 2 x y + 1$ is an identity.

Step 4

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

$f (x, y) = \int x (x y + 1) d y$

Step 5

Integrate $N (x, y) = x (x y + 1)$ to find $f (x, y)$ .

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Step 5.1

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

$f (x, y) = x \int x y + 1 d y$

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Apply the constant rule.

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

Step 5.6

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

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

Step 5.7

Simplify.

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

Step 5.8

Reorder terms.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

Differentiate using the Sum Rule.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

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

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

Step 8.2.1.2

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

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

Step 8.2.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

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

Step 8.3.7

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

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

Step 8.3.8

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

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

Step 8.3.9

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

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

Step 8.3.10

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

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

Step 8.3.11

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

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

Step 8.3.12

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

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

Step 8.3.13

Cancel the common factor of $2$ .

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Step 8.3.13.1

Cancel the common factor.

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

Step 8.3.13.2

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

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

Step 8.4

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

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

Step 8.5

Reorder terms.

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

Subtract $y$ from both sides of the equation.

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

Step 9.1.3

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

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Step 9.1.3.1

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

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

Step 9.1.3.2

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

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

Step 9.1.3.3

Add $y - x$ and $0$ .

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

Step 9.1.3.4

Subtract $y$ from $y$ .

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

Step 9.1.3.5

Add $- x$ and $0$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

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

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

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

Simplify each term.

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

Step 12.3.4

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

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

Step 12.3.5

Add $1$ and $1$ .

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

Step 12.4

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

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