Calculus Examples

$(y^{2} - 2 x y + 6) d x - (x^{2} - 2 x y + 2) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = y^{2} - 2 x y + 6$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $y^{2} - 2 x y + 6$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 2 x y] + \frac{d}{d y} [6]$ .

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 2 y - 2 x \cdot 1 + \frac{d}{d y} [6]$

Step 1.3.3

Multiply $- 2$ by $1$ .

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

Step 1.4

Since $6$ is constant with respect to $y$ , the derivative of $6$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 2 y - 2 x + 0$

Step 1.5

Simplify.

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Step 1.5.1

Add $2 y - 2 x$ and $0$ .

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

Step 1.5.2

Reorder terms.

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $- 2$ by $1$ .

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

Step 2.8

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

$\frac{\partial N}{\partial x} = - (2 x - 2 y + 0)$

Step 2.9

Add $2 x - 2 y$ and $0$ .

$\frac{\partial N}{\partial x} = - (2 x - 2 y)$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$\frac{\partial N}{\partial x} = - (2 x) - (- 2 y)$

Step 2.10.2

Combine terms.

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Step 2.10.2.1

Multiply $2$ by $- 1$ .

$\frac{\partial N}{\partial x} = - 2 x - (- 2 y)$

Step 2.10.2.2

Multiply $- 2$ by $- 1$ .

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

Step 3

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

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Step 3.1

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

$- 2 x + 2 y = - 2 x + 2 y$

Step 3.2

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

$- 2 x + 2 y = - 2 x + 2 y$ is an identity.

Step 4

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

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

Step 5

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

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Step 5.1

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 5.2

Split the single integral into multiple integrals.

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

Step 5.3

Apply the constant rule.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Apply the constant rule.

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

Step 5.7

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

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

Step 5.8

Simplify.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = y^{2} - 2 x y + 6$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

$- (y (2 x) - y^{2} \cdot 1 + \frac{d}{d x} [2 y]) + \frac{d}{d x} [g (x)] = y^{2} - 2 x y + 6$

Step 8.3.7

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

$- (y (2 x) - y^{2} \cdot 1 + 0) + \frac{d}{d x} [g (x)] = y^{2} - 2 x y + 6$

Step 8.3.8

Move $2$ to the left of $y$ .

$- (2 \cdot y x - y^{2} \cdot 1 + 0) + \frac{d}{d x} [g (x)] = y^{2} - 2 x y + 6$

Step 8.3.9

Multiply $- 1$ by $1$ .

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

Step 8.3.10

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

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

Step 8.4

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

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

Step 8.5

Simplify.

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Step 8.5.1

Apply the distributive property.

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

Step 8.5.2

Combine terms.

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Step 8.5.2.1

Multiply $2$ by $- 1$ .

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

Step 8.5.2.2

Multiply $- 1$ by $- 1$ .

$- 2 y x + 1 y^{2} + \frac{d g}{d x} = y^{2} - 2 x y + 6$

Step 8.5.2.3

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

$- 2 y x + y^{2} + \frac{d g}{d x} = y^{2} - 2 x y + 6$

Step 8.5.3

Reorder terms.

$\frac{d g}{d x} + y^{2} - 2 x y = y^{2} - 2 x y + 6$

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

$\frac{d g}{d x} - 2 x y = y^{2} - 2 x y + 6 - y^{2}$

Step 9.1.2

Add $2 x y$ to both sides of the equation.

$\frac{d g}{d x} = y^{2} - 2 x y + 6 - y^{2} + 2 x y$

Step 9.1.3

Combine the opposite terms in $y^{2} - 2 x y + 6 - y^{2} + 2 x y$ .

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Step 9.1.3.1

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

$\frac{d g}{d x} = - 2 x y + 6 + 0 + 2 x y$

Step 9.1.3.2

Add $- 2 x y + 6$ and $0$ .

$\frac{d g}{d x} = - 2 x y + 6 + 2 x y$

Step 9.1.3.3

Add $- 2 x y$ and $2 x y$ .

$\frac{d g}{d x} = 0 + 6$

Step 9.1.3.4

Add $0$ and $6$ .

$\frac{d g}{d x} = 6$

Step 10

Find the antiderivative of $6$ to find $g (x)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d x} = 6$ .

$\int \frac{d g}{d x} d x = \int 6 d x$

Step 10.2

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

$g (x) = \int 6 d x$

Step 10.3

Apply the constant rule.

$g (x) = 6 x + C$

Step 11

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

$f (x, y) = - (x^{2} y - x y^{2} + 2 y) + 6 x + C$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$f (x, y) = - (x^{2} y) - (- x y^{2}) - (2 y) + 6 x + C$

Step 12.2

Simplify.

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Step 12.2.1

Multiply $- (- x y^{2})$ .

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Step 12.2.1.1

Multiply $- 1$ by $- 1$ .

$f (x, y) = - x^{2} y + 1 (x y^{2}) - (2 y) + 6 x + C$

Step 12.2.1.2

Multiply $x$ by $1$ .

$f (x, y) = - x^{2} y + x y^{2} - (2 y) + 6 x + C$

Step 12.2.2

Multiply $2$ by $- 1$ .

$f (x, y) = - x^{2} y + x y^{2} - 2 y + 6 x + C$