Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Since $3 x^{2}$ is constant with respect to $y$ , the derivative of $3 x^{2}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [3 x y^{2}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 + 3 x \frac{d}{d y} [y^{2}]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 + 3 x (2 y)$

Step 1.3.3

Multiply $2$ by $3$ .

$\frac{\partial M}{\partial y} = 0 + 6 x y$

Step 1.4

Add $0$ and $6 x y$ .

$\frac{\partial M}{\partial y} = 6 x y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $3$ .

$\frac{\partial N}{\partial x} = 6 y x + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2 y]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- 3 y^{2}$ is constant with respect to $x$ , the derivative of $- 3 y^{2}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 6 y x + 0 + \frac{d}{d x} [2 y]$

Step 2.4.2

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

$\frac{\partial N}{\partial x} = 6 y x + 0 + 0$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Add $6 y x$ and $0$ .

$\frac{\partial N}{\partial x} = 6 y x + 0$

Step 2.5.1.2

Add $6 y x$ and $0$ .

$\frac{\partial N}{\partial x} = 6 y x$

Step 2.5.2

Reorder the factors of $6 y x$ .

$\frac{\partial N}{\partial x} = 6 x y$

Step 3

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

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Step 3.1

Substitute $6 x y$ for $\frac{\partial M}{\partial y}$ and $6 x y$ for $\frac{\partial N}{\partial x}$ .

$6 x y = 6 x y$

Step 3.2

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

$6 x y = 6 x y$ is an identity.

Step 4

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

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

Step 5

Integrate $M (x, y) = 3 x^{2} + 3 x y^{2}$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Since $3 y^{2}$ is constant with respect to $x$ , move $3 y^{2}$ out of the integral.

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

Step 5.5

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

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

Step 5.6

Simplify.

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

Step 5.7

Simplify.

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Step 5.7.1

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

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

Step 5.7.2

Cancel the common factor of $3$ .

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Step 5.7.2.1

Cancel the common factor.

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

Step 5.7.2.2

Rewrite the expression.

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

Step 5.7.3

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

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

Step 5.7.4

Combine $3$ and $\frac{1}{2}$ .

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

Step 5.7.5

Combine $y^{2}$ and $\frac{3}{2}$ .

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

Step 5.7.6

Move $3$ to the left of $y^{2}$ .

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

Step 5.7.7

Multiply $3$ by $y^{2}$ .

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

Step 5.7.8

Combine $\frac{3 y^{2}}{2}$ and $x^{2}$ .

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

Step 5.8

Reorder terms.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

Differentiate.

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Step 8.2.1

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

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

Step 8.2.2

Since $x^{3}$ is constant with respect to $y$ , the derivative of $x^{3}$ with respect to $y$ is $0$ .

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

Step 8.3

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

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Step 8.3.1

Combine $\frac{3}{2}$ and $y^{2}$ .

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

Step 8.3.2

Combine $\frac{3 y^{2}}{2}$ and $x^{2}$ .

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

Combine $2$ and $\frac{3 x^{2}}{2}$ .

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

Step 8.3.6

Multiply $3$ by $2$ .

$0 + \frac{6 x^{2}}{2} y + \frac{d}{d y} [g (y)] = 3 x^{2} y - 3 y^{2} + 2 y$

Step 8.3.7

Combine $\frac{6 x^{2}}{2}$ and $y$ .

$0 + \frac{6 x^{2} y}{2} + \frac{d}{d y} [g (y)] = 3 x^{2} y - 3 y^{2} + 2 y$

Step 8.3.8

Cancel the common factor of $6$ and $2$ .

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Step 8.3.8.1

Factor $2$ out of $6 x^{2} y$ .

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

Step 8.3.8.2

Cancel the common factors.

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Step 8.3.8.2.1

Factor $2$ out of $2$ .

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

Step 8.3.8.2.2

Cancel the common factor.

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

Step 8.3.8.2.3

Rewrite the expression.

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

Step 8.3.8.2.4

Divide $3 x^{2} y$ by $1$ .

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

Step 8.4

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

$0 + 3 x^{2} y + \frac{d g}{d y} = 3 x^{2} y - 3 y^{2} + 2 y$

Step 8.5

Simplify.

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Step 8.5.1

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

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

Step 8.5.2

Reorder terms.

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

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 $3 x^{2} y$ from both sides of the equation.

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

Step 9.1.2

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

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Step 9.1.2.1

Subtract $3 x^{2} y$ from $3 x^{2} y$ .

$\frac{d g}{d y} = - 3 y^{2} + 2 y + 0$

Step 9.1.2.2

Add $- 3 y^{2} + 2 y$ and $0$ .

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

Step 10

Find the antiderivative of $- 3 y^{2} + 2 y$ to find $g (y)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d y} = - 3 y^{2} + 2 y$ .

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

Step 10.2

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

$g (y) = \int - 3 y^{2} + 2 y d y$

Step 10.3

Split the single integral into multiple integrals.

$g (y) = \int - 3 y^{2} d y + \int 2 y d y$

Step 10.4

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

$g (y) = - 3 \int y^{2} d y + \int 2 y d y$

Step 10.5

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

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

Step 10.6

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

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

Step 10.7

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

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

Step 10.8

Simplify.

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

Step 10.9

Simplify.

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Step 10.9.1

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

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

Step 10.9.2

Cancel the common factor of $- 3$ and $3$ .

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Step 10.9.2.1

Factor $3$ out of $- 3$ .

$g (y) = \frac{3 \cdot - 1}{3} y^{3} + 2 (\frac{1}{2}) y^{2} + C$

Step 10.9.2.2

Cancel the common factors.

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Step 10.9.2.2.1

Factor $3$ out of $3$ .

$g (y) = \frac{3 \cdot - 1}{3 (1)} y^{3} + 2 (\frac{1}{2}) y^{2} + C$

Step 10.9.2.2.2

Cancel the common factor.

$g (y) = \frac{3 \cdot - 1}{3 \cdot 1} y^{3} + 2 (\frac{1}{2}) y^{2} + C$

Step 10.9.2.2.3

Rewrite the expression.

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

Step 10.9.2.2.4

Divide $- 1$ by $1$ .

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

Step 10.9.3

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

$g (y) = - y^{3} + \frac{2}{2} y^{2} + C$

Step 10.9.4

Cancel the common factor of $2$ .

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Step 10.9.4.1

Cancel the common factor.

$g (y) = - y^{3} + \frac{2}{2} y^{2} + C$

Step 10.9.4.2

Rewrite the expression.

$g (y) = - y^{3} + 1 y^{2} + C$

Step 10.9.5

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

$g (y) = - y^{3} + y^{2} + C$

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

Combine $\frac{3}{2}$ and $y^{2}$ .

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

Step 12.2

Combine $\frac{3 y^{2}}{2}$ and $x^{2}$ .

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