Calculus Examples

$(\cos (x) \sin (x) - x y^{2}) d x + y (1 - x^{2}) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

Since $\cos (x) \sin (x)$ is constant with respect to $y$ , the derivative of $\cos (x) \sin (x)$ with respect to $y$ is $0$ .

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

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 - x \frac{d}{d y} [y^{2}]$

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.4

Subtract $2 x y$ from $0$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the expression.

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Step 2.8.1

Multiply $2$ by $- 1$ .

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

Step 2.8.2

Reorder the factors of $y (- 2 x)$ .

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

Step 3

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

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Step 3.1

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

$- 2 x y = - 2 x y$

Step 3.2

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

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

Step 4

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

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

Step 5

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

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Step 5.1

Since $1 - x^{2}$ is constant with respect to $y$ , move $1 - x^{2}$ out of the integral.

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = \cos (x) \sin (x) - x y^{2}$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

Evaluate $\frac{d}{d x} [(1 - x^{2}) \frac{1}{2} y^{2}]$ .

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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

Step 8.3.5

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

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

Step 8.3.6

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

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

Step 8.3.7

Multiply $2$ by $- 1$ .

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

Step 8.3.8

Subtract $2 x$ from $0$ .

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

Step 8.3.9

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

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

Step 8.3.10

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

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

Step 8.3.11

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

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Step 8.3.11.1

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

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

Step 8.3.11.2

Cancel the common factors.

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Step 8.3.11.2.1

Factor $2$ out of $2$ .

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

Step 8.3.11.2.2

Cancel the common factor.

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

Step 8.3.11.2.3

Rewrite the expression.

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

Step 8.3.11.2.4

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

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

Step 8.4

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

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

Step 8.5

Reorder terms.

$\frac{d g}{d x} - y^{2} x = \cos (x) \sin (x) - x y^{2}$

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

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

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

Step 9.1.2

Combine the opposite terms in $\cos (x) \sin (x) - x y^{2} + y^{2} x$ .

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Step 9.1.2.1

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

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

Step 9.1.2.2

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

$\frac{d g}{d x} = \cos (x) \sin (x) + 0$

Step 9.1.2.3

Add $\cos (x) \sin (x)$ and $0$ .

$\frac{d g}{d x} = \cos (x) \sin (x)$

Step 10

Find the antiderivative of $\cos (x) \sin (x)$ to find $g (x)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d x} = \cos (x) \sin (x)$ .

$\int \frac{d g}{d x} d x = \int \cos (x) \sin (x) d x$

Step 10.2

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

$g (x) = \int \cos (x) \sin (x) d x$

Step 10.3

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- \frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.3.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

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Step 10.3.1.1

Differentiate $\cos (x)$ .

$\frac{d}{d x} [\cos (x)]$

Step 10.3.1.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- \sin (x)$

Step 10.3.2

Rewrite the problem using $u$ and $d u$ .

$g (x) = \int - u d u$

Step 10.4

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

$g (x) = - \int u d u$

Step 10.5

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

$g (x) = - (\frac{1}{2} u^{2} + C)$

Step 10.6

Rewrite $- (\frac{1}{2} u^{2} + C)$ as $- \frac{1}{2} u^{2} + C$ .

$g (x) = - \frac{1}{2} u^{2} + C$

Step 10.7

Replace all occurrences of $u$ with $\cos (x)$ .

$g (x) = - \frac{1}{2} \cos^{2} (x) + C$

Step 11

Substitute for $g (x)$ in $f (x, y) = (1 - x^{2}) \frac{1}{2} y^{2} + g (x)$ .

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

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 12.3

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

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

Step 12.4

Apply the distributive property.

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

Step 12.5

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

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

Step 12.6

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

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

Step 12.7

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

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