Calculus Examples

$(12 x y^{2} - 20 y) d x + (8 x^{2} y - 10 x) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 12 x y^{2} - 20 y$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [12 x y^{2} - 20 y]$

Step 1.2

By the Sum Rule, the derivative of $12 x y^{2} - 20 y$ with respect to $y$ is $\frac{d}{d y} [12 x y^{2}] + \frac{d}{d y} [- 20 y]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [12 x y^{2}] + \frac{d}{d y} [- 20 y]$

Step 1.3

Evaluate $\frac{d}{d y} [12 x y^{2}]$ .

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Step 1.3.1

Since $12 x$ is constant with respect to $y$ , the derivative of $12 x y^{2}$ with respect to $y$ is $12 x \frac{d}{d y} [y^{2}]$ .

$\frac{\partial M}{\partial y} = 12 x \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 20 y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 12 x (2 y) + \frac{d}{d y} [- 20 y]$

Step 1.3.3

Multiply $2$ by $12$ .

$\frac{\partial M}{\partial y} = 24 x y + \frac{d}{d y} [- 20 y]$

Step 1.4

Evaluate $\frac{d}{d y} [- 20 y]$ .

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Step 1.4.1

Since $- 20$ is constant with respect to $y$ , the derivative of $- 20 y$ with respect to $y$ is $- 20 \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 24 x y - 20 \frac{d}{d y} [y]$

Step 1.4.2

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

$\frac{\partial M}{\partial y} = 24 x y - 20 \cdot 1$

Step 1.4.3

Multiply $- 20$ by $1$ .

$\frac{\partial M}{\partial y} = 24 x y - 20$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 8 x^{2} y - 10 x$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [8 x^{2} y - 10 x]$

Step 2.2

By the Sum Rule, the derivative of $8 x^{2} y - 10 x$ with respect to $x$ is $\frac{d}{d x} [8 x^{2} y] + \frac{d}{d x} [- 10 x]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [8 x^{2} y] + \frac{d}{d x} [- 10 x]$

Step 2.3

Evaluate $\frac{d}{d x} [8 x^{2} y]$ .

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Step 2.3.1

Since $8 y$ is constant with respect to $x$ , the derivative of $8 x^{2} y$ with respect to $x$ is $8 y \frac{d}{d x} [x^{2}]$ .

$\frac{\partial N}{\partial x} = 8 y \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 8 y (2 x) + \frac{d}{d x} [- 10 x]$

Step 2.3.3

Multiply $2$ by $8$ .

$\frac{\partial N}{\partial x} = 16 y x + \frac{d}{d x} [- 10 x]$

Step 2.4

Evaluate $\frac{d}{d x} [- 10 x]$ .

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Step 2.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 16 y x - 10 \frac{d}{d x} [x]$

Step 2.4.2

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

$\frac{\partial N}{\partial x} = 16 y x - 10 \cdot 1$

Step 2.4.3

Multiply $- 10$ by $1$ .

$\frac{\partial N}{\partial x} = 16 y x - 10$

Step 2.5

Reorder terms.

$\frac{\partial N}{\partial x} = 16 x y - 10$

Step 3

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

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Step 3.1

Substitute $24 x y - 20$ for $\frac{\partial M}{\partial y}$ and $16 x y - 10$ for $\frac{\partial N}{\partial x}$ .

$24 x y - 20 = 16 x y - 10$

Step 3.2

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

$24 x y - 20 = 16 x y - 10$ 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 $24 x y - 20$ for $\frac{\partial M}{\partial y}$ .

$\frac{24 x y - 20 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $16 x y - 10$ for $\frac{\partial N}{\partial x}$ .

$\frac{24 x y - 20 - (16 x y - 10)}{N}$

Step 4.3

Substitute $8 x^{2} y - 10 x$ for $N$ .

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Step 4.3.1

Substitute $8 x^{2} y - 10 x$ for $N$ .

$\frac{24 x y - 20 - (16 x y - 10)}{8 x^{2} y - 10 x}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Apply the distributive property.

$\frac{24 x y - 20 - (16 x y) - - 10}{8 x^{2} y - 10 x}$

Step 4.3.2.2

Multiply $16$ by $- 1$ .

$\frac{24 x y - 20 - 16 (x y) - - 10}{8 x^{2} y - 10 x}$

Step 4.3.2.3

Multiply $- 1$ by $- 10$ .

$\frac{24 x y - 20 - 16 (x y) + 10}{8 x^{2} y - 10 x}$

Step 4.3.2.4

Subtract $16 x y$ from $24 x y$ .

$\frac{8 x y - 20 + 10}{8 x^{2} y - 10 x}$

Step 4.3.2.5

Add $- 20$ and $10$ .

$\frac{8 x y - 10}{8 x^{2} y - 10 x}$

Step 4.3.2.6

Factor $2$ out of $8 x y - 10$ .

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Step 4.3.2.6.1

Factor $2$ out of $8 x y$ .

$\frac{2 (4 x y) - 10}{8 x^{2} y - 10 x}$

Step 4.3.2.6.2

Factor $2$ out of $- 10$ .

$\frac{2 (4 x y) + 2 (- 5)}{8 x^{2} y - 10 x}$

Step 4.3.2.6.3

Factor $2$ out of $2 (4 x y) + 2 (- 5)$ .

$\frac{2 (4 x y - 5)}{8 x^{2} y - 10 x}$

Step 4.3.3

Factor $2 x$ out of $8 x^{2} y - 10 x$ .

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Step 4.3.3.1

Factor $2 x$ out of $8 x^{2} y$ .

$\frac{2 (4 x y - 5)}{2 x (4 x y) - 10 x}$

Step 4.3.3.2

Factor $2 x$ out of $- 10 x$ .

$\frac{2 (4 x y - 5)}{2 x (4 x y) + 2 x (- 5)}$

Step 4.3.3.3

Factor $2 x$ out of $2 x (4 x y) + 2 x (- 5)$ .

$\frac{2 (4 x y - 5)}{2 x (4 x y - 5)}$

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{2 (4 x y - 5)}{2 x (4 x y - 5)}$

Step 4.3.4.2

Rewrite the expression.

$\frac{4 x y - 5}{x (4 x y - 5)}$

Step 4.3.5

Cancel the common factor of $4 x y - 5$ .

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Step 4.3.5.1

Cancel the common factor.

$\frac{4 x y - 5}{x (4 x y - 5)}$

Step 4.3.5.2

Rewrite the expression.

$\frac{1}{x}$

Step 4.4

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

$μ (x, y) = e^{\int \frac{1}{x} d x}$

Step 5

Evaluate the integral $e^{\int \frac{1}{x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{\ln (| x |) + C}$

Step 5.2

Simplify the answer.

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Step 5.2.1

Simplify.

$μ (x, y) = e^{\ln (| x |)} + C$

Step 5.2.2

Exponentiation and log are inverse functions.

$μ (x, y) = | x | + C$

Step 6

Multiply both sides of $(12 x y^{2} - 20 y) d x + (8 x^{2} y - 10 x) d y = 0$ by the integration factor $x$ .

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Step 6.1

Multiply $12 x y^{2} - 20 y$ by $x$ .

$(12 x y^{2} - 20 y) x d x + (8 x^{2} y - 10 x) d y = 0$

Step 6.2

Apply the distributive property.

$(12 x y^{2} x - 20 y x) d x + (8 x^{2} y - 10 x) d y = 0$

Step 6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1

Move $x$ .

$(12 (x \cdot x) y^{2} - 20 y x) d x + (8 x^{2} y - 10 x) d y = 0$

Step 6.3.2

Multiply $x$ by $x$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{2} y - 10 x) d y = 0$

Step 6.4

Multiply $8 x^{2} y - 10 x$ by $x$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{2} y - 10 x) x d y = 0$

Step 6.5

Apply the distributive property.

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{2} y x - 10 x \cdot x) d y = 0$

Step 6.6

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

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Step 6.6.1

Move $x$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 (x \cdot x^{2}) y - 10 x \cdot x) d y = 0$

Step 6.6.2

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

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Step 6.6.2.1

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

$(12 x^{2} y^{2} - 20 y x) d x + (8 (x^{1} x^{2}) y - 10 x \cdot x) d y = 0$

Step 6.6.2.2

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

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{1 + 2} y - 10 x \cdot x) d y = 0$

Step 6.6.3

Add $1$ and $2$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{3} y - 10 x \cdot x) d y = 0$

Step 6.7

Multiply $x$ by $x$ by adding the exponents.

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Step 6.7.1

Move $x$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{3} y - 10 (x \cdot x)) d y = 0$

Step 6.7.2

Multiply $x$ by $x$ .

$(12 x^{2} y^{2} - 20 y x) d x + (8 x^{3} y - 10 x^{2}) d y = 0$

Step 7

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

$f (x, y) = \int 12 x^{2} y^{2} - 20 y x d x$

Step 8

Integrate $M (x, y) = 12 x^{2} y^{2} - 20 y x$ to find $f (x, y)$ .

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Step 8.1

Split the single integral into multiple integrals.

$f (x, y) = \int 12 x^{2} y^{2} d x + \int - 20 y x d x$

Step 8.2

Since $12 y^{2}$ is constant with respect to $x$ , move $12 y^{2}$ out of the integral.

$f (x, y) = 12 y^{2} \int x^{2} d x + \int - 20 y x d x$

Step 8.3

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

$f (x, y) = 12 y^{2} (\frac{1}{3} x^{3} + C) + \int - 20 y x d x$

Step 8.4

Since $- 20 y$ is constant with respect to $x$ , move $- 20 y$ out of the integral.

$f (x, y) = 12 y^{2} (\frac{1}{3} x^{3} + C) - 20 y \int x d x$

Step 8.5

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

$f (x, y) = 12 y^{2} (\frac{1}{3} x^{3} + C) - 20 y (\frac{1}{2} x^{2} + C)$

Step 8.6

Simplify.

$f (x, y) = 12 \cdot \frac{1}{3} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7

Simplify.

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Step 8.7.1

Combine $12$ and $\frac{1}{3}$ .

$f (x, y) = \frac{12}{3} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.2

Cancel the common factor of $12$ and $3$ .

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Step 8.7.2.1

Factor $3$ out of $12$ .

$f (x, y) = \frac{3 \cdot 4}{3} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.2.2

Cancel the common factors.

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Step 8.7.2.2.1

Factor $3$ out of $3$ .

$f (x, y) = \frac{3 \cdot 4}{3 (1)} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.2.2.2

Cancel the common factor.

$f (x, y) = \frac{3 \cdot 4}{3 \cdot 1} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.2.2.3

Rewrite the expression.

$f (x, y) = \frac{4}{1} y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.2.2.4

Divide $4$ by $1$ .

$f (x, y) = 4 y^{2} x^{3} - 20 y (\frac{1}{2} x^{2}) + C$

Step 8.7.3

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

$f (x, y) = 4 y^{2} x^{3} - 20 y \frac{x^{2}}{2} + C$

Step 8.7.4

Combine $- 20$ and $\frac{x^{2}}{2}$ .

$f (x, y) = 4 y^{2} x^{3} + y \frac{- 20 x^{2}}{2} + C$

Step 8.7.5

Combine $y$ and $\frac{- 20 x^{2}}{2}$ .

$f (x, y) = 4 y^{2} x^{3} + \frac{y (- 20 x^{2})}{2} + C$

Step 8.7.6

Cancel the common factor of $- 20$ and $2$ .

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Step 8.7.6.1

Factor $2$ out of $y (- 20 x^{2})$ .

$f (x, y) = 4 y^{2} x^{3} + \frac{2 (y (- 10 x^{2}))}{2} + C$

Step 8.7.6.2

Cancel the common factors.

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Step 8.7.6.2.1

Factor $2$ out of $2$ .

$f (x, y) = 4 y^{2} x^{3} + \frac{2 (y (- 10 x^{2}))}{2 (1)} + C$

Step 8.7.6.2.2

Cancel the common factor.

$f (x, y) = 4 y^{2} x^{3} + \frac{2 (y (- 10 x^{2}))}{2 \cdot 1} + C$

Step 8.7.6.2.3

Rewrite the expression.

$f (x, y) = 4 y^{2} x^{3} + \frac{y (- 10 x^{2})}{1} + C$

Step 8.7.6.2.4

Divide $y (- 10 x^{2})$ by $1$ .

$f (x, y) = 4 y^{2} x^{3} + y (- 10 x^{2}) + C$

Step 9

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

$f (x, y) = 4 y^{2} x^{3} + y (- 10 x^{2}) + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = 8 x^{3} y - 10 x^{2}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [4 y^{2} x^{3} + y (- 10 x^{2}) + g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.2

By the Sum Rule, the derivative of $4 y^{2} x^{3} + y (- 10 x^{2}) + g (y)$ with respect to $y$ is $\frac{d}{d y} [4 y^{2} x^{3}] + \frac{d}{d y} [y (- 10 x^{2})] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [4 y^{2} x^{3}] + \frac{d}{d y} [y (- 10 x^{2})] + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.3

Evaluate $\frac{d}{d y} [4 y^{2} x^{3}]$ .

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Step 11.3.1

Since $4 x^{3}$ is constant with respect to $y$ , the derivative of $4 y^{2} x^{3}$ with respect to $y$ is $4 x^{3} \frac{d}{d y} [y^{2}]$ .

$4 x^{3} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [y (- 10 x^{2})] + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.3.2

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

$4 x^{3} (2 y) + \frac{d}{d y} [y (- 10 x^{2})] + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.3.3

Multiply $2$ by $4$ .

$8 x^{3} y + \frac{d}{d y} [y (- 10 x^{2})] + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.4

Evaluate $\frac{d}{d y} [y (- 10 x^{2})]$ .

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Step 11.4.1

Since $- 10 x^{2}$ is constant with respect to $y$ , the derivative of $y (- 10 x^{2})$ with respect to $y$ is $- 10 x^{2} \frac{d}{d y} [y]$ .

$8 x^{3} y - 10 x^{2} \frac{d}{d y} [y] + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.4.2

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

$8 x^{3} y - 10 x^{2} \cdot 1 + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.4.3

Multiply $- 10$ by $1$ .

$8 x^{3} y - 10 x^{2} + \frac{d}{d y} [g (y)] = 8 x^{3} y - 10 x^{2}$

Step 11.5

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

$8 x^{3} y - 10 x^{2} + \frac{d g}{d y} = 8 x^{3} y - 10 x^{2}$

Step 11.6

Reorder terms.

$\frac{d g}{d y} - 10 x^{2} + 8 x^{3} y = 8 x^{3} y - 10 x^{2}$

Step 12

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

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Step 12.1

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

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Step 12.1.1

Add $10 x^{2}$ to both sides of the equation.

$\frac{d g}{d y} + 8 x^{3} y = 8 x^{3} y - 10 x^{2} + 10 x^{2}$

Step 12.1.2

Subtract $8 x^{3} y$ from both sides of the equation.

$\frac{d g}{d y} = 8 x^{3} y - 10 x^{2} + 10 x^{2} - 8 x^{3} y$

Step 12.1.3

Combine the opposite terms in $8 x^{3} y - 10 x^{2} + 10 x^{2} - 8 x^{3} y$ .

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Step 12.1.3.1

Add $- 10 x^{2}$ and $10 x^{2}$ .

$\frac{d g}{d y} = 8 x^{3} y + 0 - 8 x^{3} y$

Step 12.1.3.2

Add $8 x^{3} y$ and $0$ .

$\frac{d g}{d y} = 8 x^{3} y - 8 x^{3} y$

Step 12.1.3.3

Subtract $8 x^{3} y$ from $8 x^{3} y$ .

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

Step 13

Find the antiderivative of $0$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = 0$ .

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

Step 13.2

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

$g (y) = \int 0 d y$

Step 13.3

The integral of $0$ with respect to $y$ is $0$ .

$g (y) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (y) = C$

Step 14

Substitute for $g (y)$ in $f (x, y) = 4 y^{2} x^{3} + y (- 10 x^{2}) + g (y)$ .

$f (x, y) = 4 y^{2} x^{3} + y (- 10 x^{2}) + C$

Step 15

Rewrite using the commutative property of multiplication.

$f (x, y) = 4 y^{2} x^{3} - 10 y x^{2} + C$