Calculus Examples

$(x + y) d x + (x - y - 2) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [y]$

Step 1.4

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

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

Step 1.5

Add $0$ and $1$ .

$\frac{\partial M}{\partial y} = 1$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\frac{\partial N}{\partial x} = 1 + 0 + \frac{d}{d x} [- 2]$

Step 2.5

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

$\frac{\partial N}{\partial x} = 1 + 0 + 0$

Step 2.6

Combine terms.

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Step 2.6.1

Add $1$ and $0$ .

$\frac{\partial N}{\partial x} = 1 + 0$

Step 2.6.2

Add $1$ and $0$ .

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

Step 3

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

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Step 3.1

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

$1 = 1$

Step 3.2

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

$1 = 1$ is an identity.

Step 4

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

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

Step 5

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

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Step 5.1

Split the single integral into multiple integrals.

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

Step 5.2

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

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

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

Differentiate.

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Step 8.2.1

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

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

Step 8.2.2

Since $\frac{1}{2} x^{2}$ is constant with respect to $y$ , the derivative of $\frac{1}{2} x^{2}$ with respect to $y$ is $0$ .

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Multiply $x$ by $1$ .

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

Step 8.4

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

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

Step 8.5

Simplify.

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Step 8.5.1

Add $0$ and $x$ .

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

Step 8.5.2

Reorder terms.

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

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 $x$ from both sides of the equation.

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

Step 9.1.2

Combine the opposite terms in $x - y - 2 - x$ .

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Step 9.1.2.1

Subtract $x$ from $x$ .

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

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

Split the single integral into multiple integrals.

$g (y) = \int - y d y + \int - 2 d y$

Step 10.4

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

$g (y) = - \int y d y + \int - 2 d y$

Step 10.5

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

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

Step 10.6

Apply the constant rule.

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

Step 10.7

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

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

Step 10.8

Simplify.

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

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

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

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

Step 12.2

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

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