Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

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 $0$ .

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

Step 1.4.2

Add $2 x y$ and $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = y (2 x)$

Step 2.4

Reorder.

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Step 2.4.1

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

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

Step 2.4.2

Reorder the factors of $2 y x$ .

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

Step 3

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

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Step 3.1

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

$2 x y = 2 x y$

Step 3.2

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Simplify the answer.

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Step 5.3.1

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

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

Step 5.3.2

Simplify.

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.3

Reorder terms.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

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

Step 8.3.7

Cancel the common factor of $2$ .

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Step 8.3.7.1

Cancel the common factor.

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

Step 8.3.7.2

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

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

Step 8.4

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

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

Step 8.5

Reorder terms.

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

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.

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

Step 9.1.2

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

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Step 9.1.2.1

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

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

Step 9.1.2.2

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

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

Step 9.1.2.3

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

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 11

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

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

Step 12

Simplify each term.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

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

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