Calculus Examples

$(e^{y} + 6 x) d x + x e^{y} d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = e^{y} + 6 x$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [e^{y} + 6 x]$

Step 1.2

By the Sum Rule, the derivative of $e^{y} + 6 x$ with respect to $y$ is $\frac{d}{d y} [e^{y}] + \frac{d}{d y} [6 x]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [e^{y}] + \frac{d}{d y} [6 x]$

Step 1.3

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

$\frac{\partial M}{\partial y} = e^{y} + \frac{d}{d y} [6 x]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $6 x$ is constant with respect to $y$ , the derivative of $6 x$ with respect to $y$ is $0$ .

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

Step 1.4.2

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

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

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 $N (x, y)$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

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

Step 5.3

Simplify.

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = e^{y} + 6 x$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [x e^{y} + g (x)] = e^{y} + 6 x$

Step 8.2

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

$\frac{d}{d x} [x e^{y}] + \frac{d}{d x} [g (x)] = e^{y} + 6 x$

Step 8.3

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

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Step 8.3.1

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

$e^{y} \frac{d}{d x} [x] + \frac{d}{d x} [g (x)] = e^{y} + 6 x$

Step 8.3.2

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

$e^{y} \cdot 1 + \frac{d}{d x} [g (x)] = e^{y} + 6 x$

Step 8.3.3

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

$e^{y} + \frac{d}{d x} [g (x)] = e^{y} + 6 x$

Step 8.4

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

$e^{y} + \frac{d g}{d x} = e^{y} + 6 x$

Step 8.5

Reorder terms.

$\frac{d g}{d x} + e^{y} = e^{y} + 6 x$

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

$\frac{d g}{d x} = e^{y} + 6 x - e^{y}$

Step 9.1.2

Combine the opposite terms in $e^{y} + 6 x - e^{y}$ .

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Step 9.1.2.1

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

$\frac{d g}{d x} = 6 x + 0$

Step 9.1.2.2

Add $6 x$ and $0$ .

$\frac{d g}{d x} = 6 x$

Step 10

Find the antiderivative of $6 x$ to find $g (x)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d x} = 6 x$ .

$\int \frac{d g}{d x} d x = \int 6 x d x$

Step 10.2

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

$g (x) = \int 6 x d x$

Step 10.3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$g (x) = 6 \int x d x$

Step 10.4

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

$g (x) = 6 (\frac{1}{2} x^{2} + C)$

Step 10.5

Simplify the answer.

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Step 10.5.1

Rewrite $6 (\frac{1}{2} x^{2} + C)$ as $6 (\frac{1}{2}) x^{2} + C$ .

$g (x) = 6 (\frac{1}{2}) x^{2} + C$

Step 10.5.2

Simplify.

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Step 10.5.2.1

Combine $6$ and $\frac{1}{2}$ .

$g (x) = \frac{6}{2} x^{2} + C$

Step 10.5.2.2

Cancel the common factor of $6$ and $2$ .

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Step 10.5.2.2.1

Factor $2$ out of $6$ .

$g (x) = \frac{2 \cdot 3}{2} x^{2} + C$

Step 10.5.2.2.2

Cancel the common factors.

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Step 10.5.2.2.2.1

Factor $2$ out of $2$ .

$g (x) = \frac{2 \cdot 3}{2 (1)} x^{2} + C$

Step 10.5.2.2.2.2

Cancel the common factor.

$g (x) = \frac{2 \cdot 3}{2 \cdot 1} x^{2} + C$

Step 10.5.2.2.2.3

Rewrite the expression.

$g (x) = \frac{3}{1} x^{2} + C$

Step 10.5.2.2.2.4

Divide $3$ by $1$ .

$g (x) = 3 x^{2} + C$

Step 11

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

$f (x, y) = x e^{y} + 3 x^{2} + C$