Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = \sec^{2} (x) \cdot 1 + \frac{d}{d y} [\sec (x) \tan (x)]$

Step 1.3.3

Multiply $\sec^{2} (x)$ by $1$ .

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

$\frac{\partial M}{\partial y} = \sec^{2} (x) + 0$

Step 1.4.2

Add $\sec^{2} (x)$ and $0$ .

$\frac{\partial M}{\partial y} = \sec^{2} (x)$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

$\frac{\partial N}{\partial x} = \sec^{2} (x) + 0$

Step 2.4.2

Add $\sec^{2} (x)$ and $0$ .

$\frac{\partial N}{\partial x} = \sec^{2} (x)$

Step 3

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

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Step 3.1

Substitute $\sec^{2} (x)$ for $\frac{\partial M}{\partial y}$ and $\sec^{2} (x)$ for $\frac{\partial N}{\partial x}$ .

$\sec^{2} (x) = \sec^{2} (x)$

Step 3.2

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

$\sec^{2} (x) = \sec^{2} (x)$ is an identity.

Step 4

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

$f (x, y) = \int \tan (x) + 2 y d y$

Step 5

Integrate $N (x, y) = \tan (x) + 2 y$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int \tan (x) d y + \int 2 y d y$

Step 5.2

Apply the constant rule.

$f (x, y) = \tan (x) y + C + \int 2 y d y$

Step 5.3

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

$f (x, y) = \tan (x) y + C + 2 \int y d y$

Step 5.4

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

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

Step 5.5

Simplify.

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

Step 5.6

Simplify.

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Step 5.6.1

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

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

Step 5.6.2

Cancel the common factor of $2$ .

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Step 5.6.2.1

Cancel the common factor.

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

Step 5.6.2.2

Rewrite the expression.

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

Step 5.6.3

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

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = y \sec^{2} (x) + \sec (x) \tan (x)$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Simplify.

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Step 8.6.1

Add $y \sec^{2} (x)$ and $0$ .

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

Step 8.6.2

Reorder terms.

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

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 $y \sec^{2} (x)$ from both sides of the equation.

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

Step 9.1.2

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

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Step 9.1.2.1

Subtract $y \sec^{2} (x)$ from $y \sec^{2} (x)$ .

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

Step 9.1.2.2

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

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

Step 11

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

$f (x, y) = \tan (x) y + y^{2} + \sec (x) + C$

Step 12

Reorder factors in $f (x, y) = \tan (x) y + y^{2} + \sec (x) + C$ .

$f (x, y) = y \tan (x) + y^{2} + \sec (x) + C$