Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = 3 + 2 x y^{2}$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Add $0$ and $\frac{d}{d y} [2 x y^{2}]$ .

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

Multiply $2$ by $2$ .

$\frac{\partial M}{\partial y} = y (4 x y) + (3 + 2 x y^{2}) \frac{d}{d y} [y]$

Step 1.4

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

$\frac{\partial M}{\partial y} = 4 x (y^{1} y) + (3 + 2 x y^{2}) \frac{d}{d y} [y]$

Step 1.5

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

$\frac{\partial M}{\partial y} = 4 x (y^{1} y^{1}) + (3 + 2 x y^{2}) \frac{d}{d y} [y]$

Step 1.6

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

$\frac{\partial M}{\partial y} = 4 x y^{1 + 1} + (3 + 2 x y^{2}) \frac{d}{d y} [y]$

Step 1.7

Add $1$ and $1$ .

$\frac{\partial M}{\partial y} = 4 x y^{2} + (3 + 2 x y^{2}) \frac{d}{d y} [y]$

Step 1.8

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

$\frac{\partial M}{\partial y} = 4 x y^{2} + (3 + 2 x y^{2}) \cdot 1$

Step 1.9

Simplify by adding terms.

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Step 1.9.1

Multiply $3 + 2 x y^{2}$ by $1$ .

$\frac{\partial M}{\partial y} = 4 x y^{2} + 3 + 2 x y^{2}$

Step 1.9.2

Add $4 x y^{2}$ and $2 x y^{2}$ .

$\frac{\partial M}{\partial y} = 6 x y^{2} + 3$

Step 1.9.3

Reorder terms.

$\frac{\partial M}{\partial y} = 6 y^{2} x + 3$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

$\frac{\partial N}{\partial x} = 3 \frac{d}{d x} [x^{2} y^{2} + x - 1]$

Step 2.3

By the Sum Rule, the derivative of $x^{2} y^{2} + x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 3 (2 y^{2} x + 1 + 0)$

Step 2.9

Add $2 y^{2} x + 1$ and $0$ .

$\frac{\partial N}{\partial x} = 3 (2 y^{2} x + 1)$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Combine terms.

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Step 2.10.2.1

Multiply $2$ by $3$ .

$\frac{\partial N}{\partial x} = 6 (y^{2} x) + 3 \cdot 1$

Step 2.10.2.2

Multiply $3$ by $1$ .

$\frac{\partial N}{\partial x} = 6 y^{2} x + 3$

Step 3

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

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Step 3.1

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

$6 y^{2} x + 3 = 6 y^{2} x + 3$

Step 3.2

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

$6 y^{2} x + 3 = 6 y^{2} x + 3$ is an identity.

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

Apply the constant rule.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Simplify.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = 3 x + y^{2} x^{2}$ .

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

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

Step 8.3.7

Move $2$ to the left of $x^{2}$ .

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

Step 8.3.8

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

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

Step 8.3.9

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

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

Step 8.3.10

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

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

Step 8.3.11

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

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

Step 8.3.12

Add $1$ and $1$ .

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

Step 8.3.13

Multiply $3 x + y^{2} x^{2}$ by $1$ .

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

Step 8.3.14

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

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Step 8.3.14.1

Reorder $y^{2}$ and $x^{2}$ .

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

Step 8.3.14.2

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

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

Step 8.4

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

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

Step 8.5

Reorder terms.

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

Simplify $3 (x^{2} y^{2} + x - 1)$ .

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Step 9.1.1.1

Rewrite.

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

Step 9.1.1.2

Simplify by adding zeros.

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

Step 9.1.1.3

Apply the distributive property.

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

Step 9.1.1.4

Multiply $3$ by $- 1$ .

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

Step 9.1.2

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

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Step 9.1.2.1

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

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

Step 9.1.2.2

Subtract $3 x$ from both sides of the equation.

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

Step 9.1.2.3

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

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Step 9.1.2.3.1

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

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

Step 9.1.2.3.2

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

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

Step 9.1.2.3.3

Subtract $3 x$ from $3 x$ .

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

Step 9.1.2.3.4

Subtract $3$ from $0$ .

$\frac{d g}{d y} = - 3$

Step 10

Find the antiderivative of $- 3$ to find $g (y)$ .

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Step 10.1

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

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

Step 10.2

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

$g (y) = \int - 3 d y$

Step 10.3

Apply the constant rule.

$g (y) = - 3 y + C$

Step 11

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

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

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Rewrite using the commutative property of multiplication.

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

Step 12.3

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

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Step 12.3.1

Move $y^{2}$ .

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

Step 12.3.2

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

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Step 12.3.2.1

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

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

Step 12.3.2.2

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

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

Step 12.3.3

Add $2$ and $1$ .

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