Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 + 3 \cdot 1 + \frac{d}{d y} [3 y^{2}]$

Step 1.3.3

Multiply $3$ by $1$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Multiply $2$ by $3$ .

$\frac{\partial M}{\partial y} = 0 + 3 + 6 y$

Step 1.5

Simplify.

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Step 1.5.1

Add $0$ and $3$ .

$\frac{\partial M}{\partial y} = 3 + 6 y$

Step 1.5.2

Reorder terms.

$\frac{\partial M}{\partial y} = 6 y + 3$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $1$ .

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

Step 2.4

Reorder terms.

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

Step 3

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

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Step 3.1

Substitute $6 y + 3$ for $\frac{\partial M}{\partial y}$ and $2 y + 1$ for $\frac{\partial N}{\partial x}$ .

$6 y + 3 = 2 y + 1$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$6 y + 3 = 2 y + 1$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

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Step 4.1

Substitute $6 y + 3$ for $\frac{\partial M}{\partial y}$ .

$\frac{6 y + 3 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $2 y + 1$ for $\frac{\partial N}{\partial x}$ .

$\frac{6 y + 3 - (2 y + 1)}{N}$

Step 4.3

Substitute $x + 2 x y$ for $N$ .

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Step 4.3.1

Substitute $x + 2 x y$ for $N$ .

$\frac{6 y + 3 - (2 y + 1)}{x + 2 x y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{6 y + 3 - (2 y) - 1 \cdot 1}{x + 2 x y}$

Step 4.3.2.2

Multiply $2$ by $- 1$ .

$\frac{6 y + 3 - 2 y - 1 \cdot 1}{x + 2 x y}$

Step 4.3.2.3

Multiply $- 1$ by $1$ .

$\frac{6 y + 3 - 2 y - 1}{x + 2 x y}$

Step 4.3.2.4

Subtract $2 y$ from $6 y$ .

$\frac{4 y + 3 - 1}{x + 2 x y}$

Step 4.3.2.5

Subtract $1$ from $3$ .

$\frac{4 y + 2}{x + 2 x y}$

Step 4.3.2.6

Factor $2$ out of $4 y + 2$ .

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Step 4.3.2.6.1

Factor $2$ out of $4 y$ .

$\frac{2 (2 y) + 2}{x + 2 x y}$

Step 4.3.2.6.2

Factor $2$ out of $2$ .

$\frac{2 (2 y) + 2 (1)}{x + 2 x y}$

Step 4.3.2.6.3

Factor $2$ out of $2 (2 y) + 2 (1)$ .

$\frac{2 (2 y + 1)}{x + 2 x y}$

Step 4.3.3

Factor $x$ out of $x + 2 x y$ .

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Step 4.3.3.1

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

$\frac{2 (2 y + 1)}{x^{1} + 2 x y}$

Step 4.3.3.2

Factor $x$ out of $x^{1}$ .

$\frac{2 (2 y + 1)}{x \cdot 1 + 2 x y}$

Step 4.3.3.3

Factor $x$ out of $2 x y$ .

$\frac{2 (2 y + 1)}{x \cdot 1 + x (2 y)}$

Step 4.3.3.4

Factor $x$ out of $x \cdot 1 + x (2 y)$ .

$\frac{2 (2 y + 1)}{x (1 + 2 y)}$

Step 4.3.4

Cancel the common factor of $2 y + 1$ and $1 + 2 y$ .

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Step 4.3.4.1

Reorder terms.

$\frac{2 (2 y + 1)}{x (2 y + 1)}$

Step 4.3.4.2

Cancel the common factor.

$\frac{2 (2 y + 1)}{x (2 y + 1)}$

Step 4.3.4.3

Rewrite the expression.

$\frac{2}{x}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int \frac{2}{x} d x}$

Step 5

Evaluate the integral $e^{\int \frac{2}{x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{2 \int \frac{1}{x} d x}$

Step 5.2

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$μ (x, y) = e^{2 (\ln (| x |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{2 \ln (| x |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{2})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{2} + C$

Step 5.4.3

Remove the absolute value in ${| x |}^{2}$ because exponentiations with even powers are always positive.

$μ (x, y) = x^{2} + C$

Step 6

Multiply both sides of $(3 + 3 y + 3 y^{2}) d x + (x + 2 x y) d y = 0$ by the integration factor $x^{2}$ .

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $x + 2 x y$ by $x^{2}$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.5.1

Multiply $x$ by $x^{2}$ .

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Step 6.5.1.1

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

$(3 x^{2} + 3 y x^{2} + 3 y^{2} x^{2}) d x + (x^{1} x^{2} + 2 x y x^{2}) d y = 0$

Step 6.5.1.2

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

$(3 x^{2} + 3 y x^{2} + 3 y^{2} x^{2}) d x + (x^{1 + 2} + 2 x y x^{2}) d y = 0$

Step 6.5.2

Add $1$ and $2$ .

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

Step 6.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.6.1

Move $x^{2}$ .

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

Step 6.6.2

Multiply $x^{2}$ by $x$ .

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Step 6.6.2.1

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

$(3 x^{2} + 3 y x^{2} + 3 y^{2} x^{2}) d x + (x^{3} + 2 (x^{2} x^{1}) y) d y = 0$

Step 6.6.2.2

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

$(3 x^{2} + 3 y x^{2} + 3 y^{2} x^{2}) d x + (x^{3} + 2 x^{2 + 1} y) d y = 0$

Step 6.6.3

Add $2$ and $1$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

Split the single integral into multiple integrals.

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

Step 8.2

Apply the constant rule.

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Simplify.

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

Step 8.6

Simplify.

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Step 8.6.1

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

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

Step 8.6.2

Cancel the common factor of $2$ .

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Step 8.6.2.1

Cancel the common factor.

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

Step 8.6.2.2

Rewrite the expression.

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

Step 8.6.3

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

Move $3$ to the left of $y$ .

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

Step 11.4

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

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Step 11.4.1

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

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

Step 11.4.2

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

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

Step 11.4.3

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

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

Step 11.5

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

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

Step 11.6

Reorder terms.

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

Step 12

Solve for $\frac{d g}{d x}$ .

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Step 12.1

Move all terms not containing $\frac{d g}{d x}$ to the right side of the equation.

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.1.3

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

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Step 12.1.3.1

Reorder the factors in the terms $3 y x^{2}$ and $- 3 x^{2} y$ .

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

Step 12.1.3.2

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

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

Step 12.1.3.3

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

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

Step 12.1.3.4

Reorder the factors in the terms $3 y^{2} x^{2}$ and $- 3 x^{2} y^{2}$ .

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

Step 12.1.3.5

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

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

Step 12.1.3.6

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

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

$g (x) = \int 3 x^{2} d x$

Step 13.3

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

$g (x) = 3 \int x^{2} d x$

Step 13.4

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

$g (x) = 3 (\frac{1}{3} x^{3} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

Rewrite $3 (\frac{1}{3} x^{3} + C)$ as $3 (\frac{1}{3}) x^{3} + C$ .

$g (x) = 3 (\frac{1}{3}) x^{3} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

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

$g (x) = \frac{3}{3} x^{3} + C$

Step 13.5.2.2

Cancel the common factor of $3$ .

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Step 13.5.2.2.1

Cancel the common factor.

$g (x) = \frac{3}{3} x^{3} + C$

Step 13.5.2.2.2

Rewrite the expression.

$g (x) = 1 x^{3} + C$

Step 13.5.2.3

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

$g (x) = x^{3} + C$

Step 14

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

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