Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [2 y]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 + 2 \frac{d}{d y} [y]$

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $1$ .

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

Step 1.4

Add $0$ and $2$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 0 + 2 \cdot 1$

Step 2.3.3

Multiply $2$ by $1$ .

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

Step 2.4

Add $0$ and $2$ .

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

Step 3

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

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Step 3.1

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

$2 = 2$

Step 3.2

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

$2 = 2$ is an identity.

Step 4

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

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

Step 5

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

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Step 5.1

Split the single integral into multiple integrals.

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

Step 5.2

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

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

Step 5.3

Apply the constant rule.

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

Step 5.4

Simplify.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

Differentiate.

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Step 8.2.1

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

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

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 $0$ .

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Multiply $2$ by $1$ .

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

Step 8.4

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

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

Step 8.5

Simplify.

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Step 8.5.1

Add $0$ and $2 x$ .

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

Step 8.5.2

Reorder terms.

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

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 $2 x$ from both sides of the equation.

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

Step 9.1.2

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

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Step 9.1.2.1

Subtract $2 x$ from $2 x$ .

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

Step 9.1.2.2

Add $y$ and $0$ .

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

Step 10

Find the antiderivative of $y$ to find $g (y)$ .

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Step 10.1

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

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

Step 10.2

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

$g (y) = \int y d y$

Step 10.3

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

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

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

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

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

Step 12.2

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

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