Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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 left side does not equal the right side, the equation is not an identity.

$\cos (y) \sin (x) = - \cos (y) \sin (x)$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Substitute $\cos (x) \cos (y)$ for $N$ .

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Step 4.3.1

Substitute $\cos (x) \cos (y)$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $\cos (y) \sin (x)$ out of $\cos (y) \sin (x) - (- \cos (y) \sin (x))$ .

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Step 4.3.2.1.1

Factor $\cos (y) \sin (x)$ out of $\cos (y) \sin (x)$ .

$\frac{\cos (y) \sin (x) (1) - (- \cos (y) \sin (x))}{\cos (x) \cos (y)}$

Step 4.3.2.1.2

Factor $\cos (y) \sin (x)$ out of $- (- \cos (y) \sin (x))$ .

$\frac{\cos (y) \sin (x) (1) + \cos (y) \sin (x) (- (- 1))}{\cos (x) \cos (y)}$

Step 4.3.2.1.3

Factor $\cos (y) \sin (x)$ out of $\cos (y) \sin (x) (1) + \cos (y) \sin (x) (- (- 1))$ .

$\frac{\cos (y) \sin (x) (1 - (- 1))}{\cos (x) \cos (y)}$

Step 4.3.2.2

Multiply $- 1$ by $- 1$ .

$\frac{\cos (y) \sin (x) (1 + 1)}{\cos (x) \cos (y)}$

Step 4.3.2.3

Add $1$ and $1$ .

$\frac{\cos (y) \sin (x) \cdot 2}{\cos (x) \cos (y)}$

Step 4.3.3

Cancel the common factor of $\cos (y)$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{\cos (y) \sin (x) \cdot 2}{\cos (x) \cos (y)}$

Step 4.3.3.2

Rewrite the expression.

$\frac{\sin (x) \cdot 2}{\cos (x)}$

Step 4.3.4

Separate fractions.

$\frac{2}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 4.3.5

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

$\frac{2}{1} \tan (x)$

Step 4.3.6

Divide $2$ by $1$ .

$2 \tan (x)$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int 2 \tan (x) d x}$

Step 5

Evaluate the integral $e^{\int 2 \tan (x) d x}$ .

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Step 5.1

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

$μ (x, y) = e^{2 \int \tan (x) d x}$

Step 5.2

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

$μ (x, y) = e^{2 (\ln (| \sec (x) |) + C)}$

Step 5.3

Simplify.

$μ (x, y) = e^{2 \ln (| \sec (x) |)} + C$

Step 5.4

Simplify each term.

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Step 5.4.1

Simplify $2 \ln (| \sec (x) |)$ by moving $2$ inside the logarithm.

$μ (x, y) = e^{\ln ({| \sec (x) |}^{2})} + C$

Step 5.4.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| \sec (x) |}^{2} + C$

Step 5.4.3

Remove the absolute value in ${| \sec (x) |}^{2}$ because exponentiations with even powers are always positive.

$μ (x, y) = \sec^{2} (x) + C$

Step 6

Multiply both sides of $\sin (x) \sin (y) d x + \cos (x) \cos (y) d y = 0$ by the integration factor $\sec^{2} (x)$ .

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Step 6.1

Multiply $\sin (x) \sin (y)$ by $\sec^{2} (x)$ .

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

Step 6.2

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

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

Step 6.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 6.4

One to any power is one.

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

Step 6.5

Multiply $\sin (x) \sin (y) \frac{1}{\cos^{2} (x)}$ .

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.6

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 6.7

Separate fractions.

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

Step 6.8

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

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

Step 6.9

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\sin (y) \frac{1}{\cos (x)} \tan (x) d x + \cos (x) \cos (y) d y = 0$

Step 6.10

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)} \tan (x) d x + \cos (x) \cos (y) d y = 0$

Step 6.11

Simplify.

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Step 6.11.1

Divide $\sin (y)$ by $1$ .

$\sin (y) \frac{1}{\cos (x)} \tan (x) d x + \cos (x) \cos (y) d y = 0$

Step 6.11.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 6.12

Multiply $\cos (x) \cos (y)$ by $\sec^{2} (x)$ .

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

Step 6.13

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

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

Step 6.14

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 6.15

Cancel the common factor of $\cos (x)$ .

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Step 6.15.1

Factor $\cos (x)$ out of $\cos (x) \cos (y)$ .

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

Step 6.15.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 6.15.3

Cancel the common factor.

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

Step 6.15.4

Rewrite the expression.

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

Step 6.16

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

$\sin (y) \sec (x) \tan (x) d x + \frac{\cos (y) \cdot 1^{2}}{\cos (x)} d y = 0$

Step 6.17

One to any power is one.

$\sin (y) \sec (x) \tan (x) d x + \frac{\cos (y) \cdot 1}{\cos (x)} d y = 0$

Step 6.18

Multiply $\cos (y)$ by $1$ .

$\sin (y) \sec (x) \tan (x) d x + \frac{\cos (y)}{\cos (x)} d y = 0$

Step 6.19

Rewrite $\frac{\cos (y)}{\cos (x)}$ as a product.

$\sin (y) \sec (x) \tan (x) d x + \cos (y) \frac{1}{\cos (x)} d y = 0$

Step 6.20

Write $\cos (y)$ as a fraction with denominator $1$ .

$\sin (y) \sec (x) \tan (x) d x + \frac{\cos (y)}{1} \cdot \frac{1}{\cos (x)} d y = 0$

Step 6.21

Simplify.

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Step 6.21.1

Divide $\cos (y)$ by $1$ .

$\sin (y) \sec (x) \tan (x) d x + \cos (y) \frac{1}{\cos (x)} d y = 0$

Step 6.21.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Reorder terms.

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

Step 11.5.2

Simplify each term.

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Step 11.5.2.1

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

$\frac{d g}{d x} + \frac{1}{\cos (x)} \sin (y) \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 11.5.2.2

Combine $\frac{1}{\cos (x)}$ and $\sin (y)$ .

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

Step 11.5.2.3

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

$\frac{d g}{d x} + \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 11.5.2.4

Multiply $\frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 11.5.2.4.1

Multiply $\frac{\sin (y)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 11.5.2.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 11.5.2.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos^{1} (x)} = \sin (y) \sec (x) \tan (x)$

Step 11.5.2.4.4

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

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{{\cos (x)}^{1 + 1}} = \sin (y) \sec (x) \tan (x)$

Step 11.5.2.4.5

Add $1$ and $1$ .

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

Step 11.5.3

Simplify each term.

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Step 11.5.3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 11.5.3.2

Separate fractions.

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (y)}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 11.5.3.3

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)}) = \sin (y) \sec (x) \tan (x)$

Step 11.5.3.4

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)}) = \sin (y) \sec (x) \tan (x)$

Step 11.5.3.5

Simplify.

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Step 11.5.3.5.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)}) = \sin (y) \sec (x) \tan (x)$

Step 11.5.3.5.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 11.5.3.6

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

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Simplify the left side.

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Step 12.1.1.1

Simplify $\frac{d g}{d x} + \tan (x) \sin (y) \sec (x)$ .

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Step 12.1.1.1.1

Simplify each term.

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Step 12.1.1.1.1.1

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

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

Step 12.1.1.1.1.2

Combine $\frac{\sin (x)}{\cos (x)}$ and $\sin (y)$ .

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

Step 12.1.1.1.1.3

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

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.1.4

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 12.1.1.1.1.4.1

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 12.1.1.1.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos^{1} (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.1.4.4

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

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{{\cos (x)}^{1 + 1}} = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.1.4.5

Add $1$ and $1$ .

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

Step 12.1.1.1.2

Simplify each term.

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Step 12.1.1.1.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.1.1.2.2

Separate fractions.

$\frac{d g}{d x} + \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.2.3

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

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

Step 12.1.1.1.2.4

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \sin (y) \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.2.5

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.2.6

Simplify.

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Step 12.1.1.1.2.6.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \sin (y) \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.1.1.2.6.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.2

Simplify the right side.

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Step 12.1.2.1

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

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Step 12.1.2.1.1

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \sin (y) \frac{1}{\cos (x)} \tan (x)$

Step 12.1.2.1.2

Combine $\sin (y)$ and $\frac{1}{\cos (x)}$ .

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

Step 12.1.2.1.3

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 12.1.2.1.4

Multiply $\frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 12.1.2.1.4.1

Multiply $\frac{\sin (y)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 12.1.2.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos (x)}$

Step 12.1.2.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos^{1} (x)}$

Step 12.1.2.1.4.4

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{{\cos (x)}^{1 + 1}}$

Step 12.1.2.1.4.5

Add $1$ and $1$ .

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

Step 12.1.2.1.5

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.2.1.6

Separate fractions.

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (y)}{\cos (x)}$

Step 12.1.2.1.7

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)})$

Step 12.1.2.1.8

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)})$

Step 12.1.2.1.9

Simplify.

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Step 12.1.2.1.9.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)})$

Step 12.1.2.1.9.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.2.1.10

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

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

Step 12.1.3

Simplify $\frac{d g}{d x} + \sin (y) \sec (x) \tan (x)$ .

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Step 12.1.3.1

Simplify each term.

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Step 12.1.3.1.1

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

$\frac{d g}{d x} + \sin (y) \frac{1}{\cos (x)} \tan (x) = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.1.2

Combine $\sin (y)$ and $\frac{1}{\cos (x)}$ .

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

Step 12.1.3.1.3

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

$\frac{d g}{d x} + \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.1.4

Multiply $\frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 12.1.3.1.4.1

Multiply $\frac{\sin (y)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 12.1.3.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos (x)} = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos^{1} (x)} = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.1.4.4

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

$\frac{d g}{d x} + \frac{\sin (y) \sin (x)}{{\cos (x)}^{1 + 1}} = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.1.4.5

Add $1$ and $1$ .

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

Step 12.1.3.2

Simplify each term.

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Step 12.1.3.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.3.2.2

Separate fractions.

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (y)}{\cos (x)} = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.2.3

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)}) = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.2.4

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)}) = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.2.5

Simplify.

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Step 12.1.3.2.5.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)}) = \tan (x) \sin (y) \sec (x)$

Step 12.1.3.2.5.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.3.2.6

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

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

Step 12.1.4

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

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Step 12.1.4.1

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

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

Step 12.1.4.2

Combine $\frac{\sin (x)}{\cos (x)}$ and $\sin (y)$ .

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

Step 12.1.4.3

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

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)}$

Step 12.1.4.4

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 12.1.4.4.1

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 12.1.4.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos (x)}$

Step 12.1.4.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos^{1} (x)}$

Step 12.1.4.4.4

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

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (x) \sin (y)}{{\cos (x)}^{1 + 1}}$

Step 12.1.4.4.5

Add $1$ and $1$ .

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

Step 12.1.4.5

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.4.6

Separate fractions.

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 12.1.4.7

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

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

Step 12.1.4.8

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \sin (y) \frac{1}{\cos (x)} \tan (x)$

Step 12.1.4.9

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)} \tan (x)$

Step 12.1.4.10

Simplify.

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Step 12.1.4.10.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \tan (x) \sin (y) \sec (x) = \sin (y) \frac{1}{\cos (x)} \tan (x)$

Step 12.1.4.10.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.5

Simplify $\frac{d g}{d x} + \tan (x) \sin (y) \sec (x)$ .

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Step 12.1.5.1

Simplify each term.

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Step 12.1.5.1.1

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

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

Step 12.1.5.1.2

Combine $\frac{\sin (x)}{\cos (x)}$ and $\sin (y)$ .

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

Step 12.1.5.1.3

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

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.1.4

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Step 12.1.5.1.4.1

Multiply $\frac{\sin (x) \sin (y)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

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

Step 12.1.5.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{\cos^{1} (x) \cos^{1} (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.1.4.4

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

$\frac{d g}{d x} + \frac{\sin (x) \sin (y)}{{\cos (x)}^{1 + 1}} = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.1.4.5

Add $1$ and $1$ .

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

Step 12.1.5.2

Simplify each term.

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Step 12.1.5.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.5.2.2

Separate fractions.

$\frac{d g}{d x} + \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.2.3

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

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

Step 12.1.5.2.4

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \sin (y) \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.2.5

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.2.6

Simplify.

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Step 12.1.5.2.6.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \sin (y) \frac{1}{\cos (x)} \tan (x) = \sin (y) \sec (x) \tan (x)$

Step 12.1.5.2.6.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.6

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

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Step 12.1.6.1

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \sin (y) \frac{1}{\cos (x)} \tan (x)$

Step 12.1.6.2

Combine $\sin (y)$ and $\frac{1}{\cos (x)}$ .

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

Step 12.1.6.3

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 12.1.6.4

Multiply $\frac{\sin (y)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 12.1.6.4.1

Multiply $\frac{\sin (y)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 12.1.6.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos (x)}$

Step 12.1.6.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{\cos^{1} (x) \cos^{1} (x)}$

Step 12.1.6.4.4

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

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (y) \sin (x)}{{\cos (x)}^{1 + 1}}$

Step 12.1.6.4.5

Add $1$ and $1$ .

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

Step 12.1.6.5

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

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

Step 12.1.6.6

Separate fractions.

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (y)}{\cos (x)}$

Step 12.1.6.7

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)})$

Step 12.1.6.8

Write $\sin (y)$ as a fraction with denominator $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)})$

Step 12.1.6.9

Simplify.

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Step 12.1.6.9.1

Divide $\sin (y)$ by $1$ .

$\frac{d g}{d x} + \sin (y) \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x)} (\sin (y) \frac{1}{\cos (x)})$

Step 12.1.6.9.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 12.1.6.10

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

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

Step 12.1.7

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

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Step 12.1.7.1

Subtract $\sin (y) \sec (x) \tan (x)$ from both sides of the equation.

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

Step 12.1.7.2

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

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Step 12.1.7.2.1

Reorder the factors in the terms $\tan (x) \sin (y) \sec (x)$ and $- \sin (y) \sec (x) \tan (x)$ .

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

Step 12.1.7.2.2

Subtract $\sec (x) \sin (y) \tan (x)$ from $\sec (x) \sin (y) \tan (x)$ .

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

Step 13

Find the antiderivative of $0$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 0$ .

$\int \frac{d g}{d x} d x = \int 0 d x$

Step 13.2

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

$g (x) = \int 0 d x$

Step 13.3

The integral of $0$ with respect to $x$ is $0$ .

$g (x) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (x) = C$

Step 14

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

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

Step 15

Simplify $f (x, y) = \sec (x) \sin (y) + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

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

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

Step 15.1.2

Combine $\frac{1}{\cos (x)}$ and $\sin (y)$ .

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

Step 15.2

Simplify each term.

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Step 15.2.1

Rewrite $\frac{\sin (y)}{\cos (x)}$ as a product.

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

Step 15.2.2

Write $\sin (y)$ as a fraction with denominator $1$ .

$f (x, y) = \frac{\sin (y)}{1} \cdot \frac{1}{\cos (x)} + C$

Step 15.2.3

Simplify.

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Step 15.2.3.1

Divide $\sin (y)$ by $1$ .

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

Step 15.2.3.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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