Calculus Examples

$(\tan (x) - \sin (x) \sin (y)) d x + (\cos (x) \cos (y)) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [- \sin (x) \sin (y)]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 - \sin (x) \frac{d}{d y} [\sin (y)]$

Step 1.3.2

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

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

Step 1.4

Simplify.

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Step 1.4.1

Subtract $\sin (x) \cos (y)$ from $0$ .

$\frac{\partial M}{\partial y} = - \sin (x) \cos (y)$

Step 1.4.2

Reorder the factors of $- \sin (x) \cos (y)$ .

$\frac{\partial M}{\partial y} = - \cos (y) \sin (x)$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = \cos (x) \cos (y)$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [\cos (x) \cos (y)]$

Step 2.2

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

$\frac{\partial N}{\partial x} = \cos (y) \frac{d}{d x} [\cos (x)]$

Step 2.3

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

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

Step 2.4

Reorder the factors of $\cos (y) (- \sin (x))$ .

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

Step 3

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

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Step 3.1

Substitute $- \cos (y) \sin (x)$ for $\frac{\partial M}{\partial y}$ and $- \cos (y) \sin (x)$ for $\frac{\partial N}{\partial x}$ .

$- \cos (y) \sin (x) = - \cos (y) \sin (x)$

Step 3.2

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

$- \cos (y) \sin (x) = - \cos (y) \sin (x)$ is an identity.

Step 4

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

$f (x, y) = \int \cos (x) \cos (y) d y$

Step 5

Integrate $N (x, y) = \cos (x) \cos (y)$ to find $f (x, y)$ .

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Step 5.1

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

$f (x, y) = \cos (x) \int \cos (y) d y$

Step 5.2

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

$f (x, y) = \cos (x) (\sin (y) + C)$

Step 5.3

Simplify.

$f (x, y) = \cos (x) \sin (y) + C$

Step 6

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

$f (x, y) = \cos (x) \sin (y) + g (x)$

Step 7

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

$\frac{\partial f}{\partial x} = \tan (x) - \sin (x) \sin (y)$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

Evaluate $\frac{d}{d x} [\cos (x) \sin (y)]$ .

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.4

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

$\sin (y) (- \sin (x)) + \frac{d g}{d x} = \tan (x) - \sin (x) \sin (y)$

Step 8.5

Reorder terms.

$\frac{d g}{d x} - \sin (x) \sin (y) = \tan (x) - \sin (x) \sin (y)$

Step 9

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

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Step 9.1

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

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Step 9.1.1

Simplify the right side.

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Step 9.1.1.1

Simplify $\tan (x) - \sin (x) \sin (y)$ .

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Step 9.1.1.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 9.1.1.1.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d g}{d x} - \sin (x) \sin (y) = \tan (x) - \sin (x) \sin (y)$

Step 9.1.2

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

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Step 9.1.2.1

Add $\sin (x) \sin (y)$ to both sides of the equation.

$\frac{d g}{d x} = \tan (x) - \sin (x) \sin (y) + \sin (x) \sin (y)$

Step 9.1.2.2

Combine the opposite terms in $\tan (x) - \sin (x) \sin (y) + \sin (x) \sin (y)$ .

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Step 9.1.2.2.1

Add $- \sin (x) \sin (y)$ and $\sin (x) \sin (y)$ .

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

Step 9.1.2.2.2

Add $\tan (x)$ and $0$ .

$\frac{d g}{d x} = \tan (x)$

Step 9.1.2.3

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 9.1.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d g}{d x} = \tan (x)$

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$g (x) = \ln (| \sec (x) |) + C$

Step 11

Substitute for $g (x)$ in $f (x, y) = \cos (x) \sin (y) + g (x)$ .

$f (x, y) = \cos (x) \sin (y) + \ln (| \sec (x) |) + C$