Calculus Examples

$3 x (x y - 2) d x + (x^{3} + 2 y) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 x (x y - 2)$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [3 x (x y - 2)]$

Step 1.2

Since $3 x$ is constant with respect to $y$ , the derivative of $3 x (x y - 2)$ with respect to $y$ is $3 x \frac{d}{d y} [x y - 2]$ .

$\frac{\partial M}{\partial y} = 3 x \frac{d}{d y} [x y - 2]$

Step 1.3

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

$\frac{\partial M}{\partial y} = 3 x (\frac{d}{d y} [x y] + \frac{d}{d y} [- 2])$

Step 1.4

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

$\frac{\partial M}{\partial y} = 3 x (x \frac{d}{d y} [y] + \frac{d}{d y} [- 2])$

Step 1.5

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

$\frac{\partial M}{\partial y} = 3 x (x \cdot 1 + \frac{d}{d y} [- 2])$

Step 1.6

Multiply $x$ by $1$ .

$\frac{\partial M}{\partial y} = 3 x (x + \frac{d}{d y} [- 2])$

Step 1.7

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

$\frac{\partial M}{\partial y} = 3 x (x + 0)$

Step 1.8

Add $x$ and $0$ .

$\frac{\partial M}{\partial y} = 3 x \cdot x$

Step 1.9

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

$\frac{\partial M}{\partial y} = 3 (x^{1} x)$

Step 1.10

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

$\frac{\partial M}{\partial y} = 3 (x^{1} x^{1})$

Step 1.11

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

$\frac{\partial M}{\partial y} = 3 x^{1 + 1}$

Step 1.12

Add $1$ and $1$ .

$\frac{\partial M}{\partial y} = 3 x^{2}$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x^{3} + 2 y$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{3} + 2 y]$

Step 2.2

By the Sum Rule, the derivative of $x^{3} + 2 y$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 y]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2 y]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{\partial N}{\partial x} = 3 x^{2} + \frac{d}{d x} [2 y]$

Step 2.4

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

$\frac{\partial N}{\partial x} = 3 x^{2} + 0$

Step 2.5

Add $3 x^{2}$ and $0$ .

$\frac{\partial N}{\partial x} = 3 x^{2}$

Step 3

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

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Step 3.1

Substitute $3 x^{2}$ for $\frac{\partial M}{\partial y}$ and $3 x^{2}$ for $\frac{\partial N}{\partial x}$ .

$3 x^{2} = 3 x^{2}$

Step 3.2

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

$3 x^{2} = 3 x^{2}$ is an identity.

Step 4

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

$f (x, y) = \int 3 x (x y - 2) d x$

Step 5

Integrate $M (x, y) = 3 x (x y - 2)$ to find $f (x, y)$ .

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Step 5.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$f (x, y) = 3 \int x (x y - 2) d x$

Step 5.2

Expand $x (x y - 2)$ .

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Step 5.2.1

Apply the distributive property.

$f (x, y) = 3 \int x (x y) + x \cdot - 2 d x$

Step 5.2.2

Remove parentheses.

$f (x, y) = 3 \int x \cdot x y + x \cdot - 2 d x$

Step 5.2.3

Reorder $x$ and $- 2$ .

$f (x, y) = 3 \int x \cdot x y - 2 x d x$

Step 5.2.4

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

$f (x, y) = 3 \int x^{1} x y - 2 x d x$

Step 5.2.5

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

$f (x, y) = 3 \int x^{1} x^{1} y - 2 x d x$

Step 5.2.6

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

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

Step 5.2.7

Add $1$ and $1$ .

$f (x, y) = 3 \int x^{2} y - 2 x d x$

Step 5.3

Split the single integral into multiple integrals.

$f (x, y) = 3 (\int x^{2} y d x + \int - 2 x d x)$

Step 5.4

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

$f (x, y) = 3 (y \int x^{2} d x + \int - 2 x d x)$

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Simplify.

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

Step 5.9

Reorder terms.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial y} = x^{3} + 2 y$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [3 (\frac{1}{3} y x^{3} - x^{2}) + g (y)] = x^{3} + 2 y$

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

$\frac{d}{d y} [3 (\frac{y}{3} x^{3} - x^{2}) + g (y)] = x^{3} + 2 y$

Step 8.2.1.2

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

$\frac{d}{d y} [3 (\frac{y x^{3}}{3} - x^{2}) + g (y)] = x^{3} + 2 y$

Step 8.2.2

By the Sum Rule, the derivative of $3 (\frac{y x^{3}}{3} - x^{2}) + g (y)$ with respect to $y$ is $\frac{d}{d y} [3 (\frac{y x^{3}}{3} - x^{2})] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [3 (\frac{y x^{3}}{3} - x^{2})] + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3

Evaluate $\frac{d}{d y} [3 (\frac{y x^{3}}{3} - x^{2})]$ .

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Step 8.3.1

Since $3$ is constant with respect to $y$ , the derivative of $3 (\frac{y x^{3}}{3} - x^{2})$ with respect to $y$ is $3 \frac{d}{d y} [\frac{y x^{3}}{3} - x^{2}]$ .

$3 \frac{d}{d y} [\frac{y x^{3}}{3} - x^{2}] + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.2

By the Sum Rule, the derivative of $\frac{y x^{3}}{3} - x^{2}$ with respect to $y$ is $\frac{d}{d y} [\frac{y x^{3}}{3}] + \frac{d}{d y} [- x^{2}]$ .

$3 (\frac{d}{d y} [\frac{y x^{3}}{3}] + \frac{d}{d y} [- x^{2}]) + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.3

Since $\frac{x^{3}}{3}$ is constant with respect to $y$ , the derivative of $\frac{y x^{3}}{3}$ with respect to $y$ is $\frac{x^{3}}{3} \frac{d}{d y} [y]$ .

$3 (\frac{x^{3}}{3} \frac{d}{d y} [y] + \frac{d}{d y} [- x^{2}]) + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.4

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

$3 (\frac{x^{3}}{3} \cdot 1 + \frac{d}{d y} [- x^{2}]) + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.5

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

$3 (\frac{x^{3}}{3} \cdot 1 + 0) + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.6

Multiply $\frac{x^{3}}{3}$ by $1$ .

$3 (\frac{x^{3}}{3} + 0) + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.7

Add $\frac{x^{3}}{3}$ and $0$ .

$3 \frac{x^{3}}{3} + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.8

Combine $3$ and $\frac{x^{3}}{3}$ .

$\frac{3 x^{3}}{3} + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.9

Cancel the common factor of $3$ .

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Step 8.3.9.1

Cancel the common factor.

$\frac{3 x^{3}}{3} + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.3.9.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d y} [g (y)] = x^{3} + 2 y$

Step 8.4

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

$x^{3} + \frac{d g}{d y} = x^{3} + 2 y$

Step 8.5

Reorder terms.

$\frac{d g}{d y} + x^{3} = x^{3} + 2 y$

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 $x^{3}$ from both sides of the equation.

$\frac{d g}{d y} = x^{3} + 2 y - x^{3}$

Step 9.1.2

Combine the opposite terms in $x^{3} + 2 y - x^{3}$ .

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Step 9.1.2.1

Subtract $x^{3}$ from $x^{3}$ .

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

Step 9.1.2.2

Add $2 y$ and $0$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Simplify the answer.

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Step 10.5.1

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

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

Step 10.5.2

Simplify.

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Step 10.5.2.1

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

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

Step 10.5.2.2

Cancel the common factor of $2$ .

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Step 10.5.2.2.1

Cancel the common factor.

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

Step 10.5.2.2.2

Rewrite the expression.

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

Step 10.5.2.3

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

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

Step 11

Substitute for $g (y)$ in $f (x, y) = 3 (\frac{1}{3} y x^{3} - x^{2}) + g (y)$ .

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

Step 12

Simplify each term.

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Step 12.1

Simplify each term.

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Cancel the common factor of $3$ .

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Step 12.3.1

Cancel the common factor.

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

Step 12.3.2

Rewrite the expression.

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

Step 12.4

Multiply $- 1$ by $3$ .

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