Calculus Examples

$e^{y} d x + (\cos (y) + x e^{y}) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [e^{y}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d y} [a^{y}]$ is $a^{y} \ln (a)$ where $a$ = $e$ .

$\frac{\partial M}{\partial y} = e^{y}$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

$\frac{\partial N}{\partial x} = 0 + \frac{d}{d x} [x e^{y}]$

Step 2.3

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

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Step 2.3.1

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

$\frac{\partial N}{\partial x} = 0 + e^{y} \frac{d}{d x} [x]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 0 + e^{y} \cdot 1$

Step 2.3.3

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

$\frac{\partial N}{\partial x} = 0 + e^{y}$

Step 2.4

Add $0$ and $e^{y}$ .

$\frac{\partial N}{\partial x} = e^{y}$

Step 3

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

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Step 3.1

Substitute $e^{y}$ for $\frac{\partial M}{\partial y}$ and $e^{y}$ for $\frac{\partial N}{\partial x}$ .

$e^{y} = e^{y}$

Step 3.2

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

$e^{y} = e^{y}$ is an identity.

Step 4

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

$f (x, y) = \int e^{y} d x$

Step 5

Integrate $M (x, y) = e^{y}$ to find $f (x, y)$ .

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Step 5.1

Apply the constant rule.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial y} = \cos (y) + x e^{y}$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d y} [a^{y}]$ is $a^{y} \ln (a)$ where $a$ = $e$ .

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

Step 8.4

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

$x e^{y} + \frac{d g}{d y} = \cos (y) + x e^{y}$

Step 8.5

Simplify.

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Step 8.5.1

Reorder terms.

$\frac{d g}{d y} + e^{y} x = \cos (y) + x e^{y}$

Step 8.5.2

Reorder factors in $\frac{d g}{d y} + e^{y} x$ .

$\frac{d g}{d y} + x e^{y} = \cos (y) + x e^{y}$

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

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

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Step 9.1.2.1

Subtract $x e^{y}$ from $x e^{y}$ .

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

Step 9.1.2.2

Add $\cos (y)$ and $0$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

$g (y) = \int \cos (y) d y$

Step 10.3

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

$g (y) = \sin (y) + C$

Step 11

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

$f (x, y) = e^{y} x + \sin (y) + C$

Step 12

Reorder factors in $f (x, y) = e^{y} x + \sin (y) + C$ .

$f (x, y) = x e^{y} + \sin (y) + C$