Calculus Examples

$3 y (x^{2} - 1) d x + (x^{3} + 8 y - 3 x) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 3 y (x^{2} - 1)$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Since $3 (x^{2} - 1)$ is constant with respect to $y$ , the derivative of $3 y (x^{2} - 1)$ with respect to $y$ is $3 (x^{2} - 1) \frac{d}{d y} [y]$ .

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

Step 1.3

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

$\frac{\partial M}{\partial y} = 3 (x^{2} - 1) \cdot 1$

Step 1.4

Multiply $3$ by $1$ .

$\frac{\partial M}{\partial y} = 3 (x^{2} - 1)$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$\frac{\partial M}{\partial y} = 3 x^{2} + 3 \cdot - 1$

Step 1.5.2

Multiply $3$ by $- 1$ .

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x^{3} + 8 y - 3 x$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{3} + 8 y - 3 x]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{3} + 8 y - 3 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 3 x]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 3 x]$

Step 2.2.2

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

$\frac{\partial N}{\partial x} = 3 x^{2} + \frac{d}{d x} [8 y] + \frac{d}{d x} [- 3 x]$

Step 2.2.3

Since $8 y$ is constant with respect to $x$ , the derivative of $8 y$ with respect to $x$ is $0$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 3 x^{2} + 0 - 3 \cdot 1$

Step 2.3.3

Multiply $- 3$ by $1$ .

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

Step 2.4

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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

Step 5

Integrate $M (x, y) = 3 y (x^{2} - 1)$ to find $f (x, y)$ .

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Step 5.1

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

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

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

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

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

Step 5.4

Apply the constant rule.

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

Step 5.5

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

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

Step 5.6

Simplify.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

Differentiate using the Sum Rule.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Multiply $3$ by $1$ .

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

Step 8.4

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

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

Step 8.5

Simplify.

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Step 8.5.1

Apply the distributive property.

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

Step 8.5.2

Combine terms.

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Step 8.5.2.1

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

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

Step 8.5.2.2

Cancel the common factor of $3$ .

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Step 8.5.2.2.1

Cancel the common factor.

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

Step 8.5.2.2.2

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

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

Step 8.5.2.3

Multiply $- 1$ by $3$ .

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

Step 8.5.3

Reorder terms.

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

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.

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

Step 9.1.2

Add $3 x$ to both sides of the equation.

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

Step 9.1.3

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

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Step 9.1.3.1

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

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

Step 9.1.3.2

Add $8 y - 3 x$ and $0$ .

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

Step 9.1.3.3

Add $- 3 x$ and $3 x$ .

$\frac{d g}{d y} = 8 y + 0$

Step 9.1.3.4

Add $8 y$ and $0$ .

$\frac{d g}{d y} = 8 y$

Step 10

Find the antiderivative of $8 y$ to find $g (y)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d y} = 8 y$ .

$\int \frac{d g}{d y} d y = \int 8 y d y$

Step 10.2

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

$g (y) = \int 8 y d y$

Step 10.3

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$g (y) = 8 \int y d y$

Step 10.4

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

$g (y) = 8 (\frac{1}{2} y^{2} + C)$

Step 10.5

Simplify the answer.

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Step 10.5.1

Rewrite $8 (\frac{1}{2} y^{2} + C)$ as $8 (\frac{1}{2}) y^{2} + C$ .

$g (y) = 8 (\frac{1}{2}) y^{2} + C$

Step 10.5.2

Simplify.

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Step 10.5.2.1

Combine $8$ and $\frac{1}{2}$ .

$g (y) = \frac{8}{2} y^{2} + C$

Step 10.5.2.2

Cancel the common factor of $8$ and $2$ .

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Step 10.5.2.2.1

Factor $2$ out of $8$ .

$g (y) = \frac{2 \cdot 4}{2} y^{2} + C$

Step 10.5.2.2.2

Cancel the common factors.

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Step 10.5.2.2.2.1

Factor $2$ out of $2$ .

$g (y) = \frac{2 \cdot 4}{2 (1)} y^{2} + C$

Step 10.5.2.2.2.2

Cancel the common factor.

$g (y) = \frac{2 \cdot 4}{2 \cdot 1} y^{2} + C$

Step 10.5.2.2.2.3

Rewrite the expression.

$g (y) = \frac{4}{1} y^{2} + C$

Step 10.5.2.2.2.4

Divide $4$ by $1$ .

$g (y) = 4 y^{2} + C$

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Cancel the common factor of $3$ .

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Step 12.3.1

Factor $3$ out of $3 y$ .

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

Step 12.3.2

Cancel the common factor.

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

Step 12.3.3

Rewrite the expression.

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

Step 12.4

Rewrite using the commutative property of multiplication.

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

Step 12.5

Multiply $3$ by $- 1$ .

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