Calculus Examples

$(2 x \sin (y) + y^{3} e^{x}) d x + (x^{2} \cos (y) + 3 y^{2} e^{x}) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 2 x \sin (y) + y^{3} e^{x}$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 x \sin (y) + y^{3} e^{x}]$

Step 1.2

By the Sum Rule, the derivative of $2 x \sin (y) + y^{3} e^{x}$ with respect to $y$ is $\frac{d}{d y} [2 x \sin (y)] + \frac{d}{d y} [y^{3} e^{x}]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [2 x \sin (y)] + \frac{d}{d y} [y^{3} e^{x}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 2 x \frac{d}{d y} [\sin (y)] + \frac{d}{d y} [y^{3} e^{x}]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 2 x \cos (y) + \frac{d}{d y} [y^{3} e^{x}]$

Step 1.4

Evaluate $\frac{d}{d y} [y^{3} e^{x}]$ .

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Step 1.4.1

Since $e^{x}$ is constant with respect to $y$ , the derivative of $y^{3} e^{x}$ with respect to $y$ is $e^{x} \frac{d}{d y} [y^{3}]$ .

$\frac{\partial M}{\partial y} = 2 x \cos (y) + e^{x} \frac{d}{d y} [y^{3}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$\frac{\partial M}{\partial y} = 2 x \cos (y) + e^{x} (3 y^{2})$

Step 1.4.3

Move $3$ to the left of $e^{x}$ .

$\frac{\partial M}{\partial y} = 2 x \cos (y) + 3 e^{x} y^{2}$

Step 1.5

Reorder factors in $2 x \cos (y) + 3 e^{x} y^{2}$ .

$\frac{\partial M}{\partial y} = 2 x \cos (y) + 3 y^{2} e^{x}$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x^{2} \cos (y) + 3 y^{2} e^{x}$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} \cos (y) + 3 y^{2} e^{x}]$

Step 2.2

By the Sum Rule, the derivative of $x^{2} \cos (y) + 3 y^{2} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} \cos (y)] + \frac{d}{d x} [3 y^{2} e^{x}]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [x^{2} \cos (y)] + \frac{d}{d x} [3 y^{2} e^{x}]$

Step 2.3

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

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Step 2.3.1

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

$\frac{\partial N}{\partial x} = \cos (y) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 y^{2} e^{x}]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{\partial N}{\partial x} = \cos (y) (2 x) + \frac{d}{d x} [3 y^{2} e^{x}]$

Step 2.3.3

Move $2$ to the left of $\cos (y)$ .

$\frac{\partial N}{\partial x} = 2 \cos (y) x + \frac{d}{d x} [3 y^{2} e^{x}]$

Step 2.4

Evaluate $\frac{d}{d x} [3 y^{2} e^{x}]$ .

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Step 2.4.1

Since $3 y^{2}$ is constant with respect to $x$ , the derivative of $3 y^{2} e^{x}$ with respect to $x$ is $3 y^{2} \frac{d}{d x} [e^{x}]$ .

$\frac{\partial N}{\partial x} = 2 \cos (y) x + 3 y^{2} \frac{d}{d x} [e^{x}]$

Step 2.4.2

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

$\frac{\partial N}{\partial x} = 2 \cos (y) x + 3 y^{2} e^{x}$

Step 2.5

Simplify.

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Step 2.5.1

Reorder terms.

$\frac{\partial N}{\partial x} = 2 x \cos (y) + 3 e^{x} y^{2}$

Step 2.5.2

Reorder factors in $2 x \cos (y) + 3 e^{x} y^{2}$ .

$\frac{\partial N}{\partial x} = 2 x \cos (y) + 3 y^{2} e^{x}$

Step 3

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

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Step 3.1

Substitute $2 x \cos (y) + 3 y^{2} e^{x}$ for $\frac{\partial M}{\partial y}$ and $2 x \cos (y) + 3 y^{2} e^{x}$ for $\frac{\partial N}{\partial x}$ .

$2 x \cos (y) + 3 y^{2} e^{x} = 2 x \cos (y) + 3 y^{2} e^{x}$

Step 3.2

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

$2 x \cos (y) + 3 y^{2} e^{x} = 2 x \cos (y) + 3 y^{2} e^{x}$ is an identity.

Step 4

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

$f (x, y) = \int x^{2} \cos (y) + 3 y^{2} e^{x} d y$

Step 5

Integrate $N (x, y) = x^{2} \cos (y) + 3 y^{2} e^{x}$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int x^{2} \cos (y) d y + \int 3 y^{2} e^{x} d y$

Step 5.2

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

$f (x, y) = x^{2} \int \cos (y) d y + \int 3 y^{2} e^{x} d y$

Step 5.3

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

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

Step 5.4

Since $3 e^{x}$ is constant with respect to $y$ , move $3 e^{x}$ out of the integral.

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

Step 5.5

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$f (x, y) = x^{2} (\sin (y) + C) + 3 e^{x} (\frac{1}{3} y^{3} + C)$

Step 5.6

Simplify.

$f (x, y) = x^{2} \sin (y) + 3 \cdot \frac{1}{3} e^{x} y^{3} + C$

Step 5.7

Simplify.

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Step 5.7.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 5.7.2

Cancel the common factor of $3$ .

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Step 5.7.2.1

Cancel the common factor.

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

Step 5.7.2.2

Rewrite the expression.

$f (x, y) = x^{2} \sin (y) + 1 e^{x} y^{3} + C$

Step 5.7.3

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

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

Step 6

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

$f (x, y) = x^{2} \sin (y) + e^{x} y^{3} + g (x)$

Step 7

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

$\frac{\partial f}{\partial x} = 2 x \sin (y) + y^{3} e^{x}$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [x^{2} \sin (y) + e^{x} y^{3} + g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.2

By the Sum Rule, the derivative of $x^{2} \sin (y) + e^{x} y^{3} + g (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2} \sin (y)] + \frac{d}{d x} [e^{x} y^{3}] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [x^{2} \sin (y)] + \frac{d}{d x} [e^{x} y^{3}] + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.3

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

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Step 8.3.1

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

$\sin (y) \frac{d}{d x} [x^{2}] + \frac{d}{d x} [e^{x} y^{3}] + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\sin (y) (2 x) + \frac{d}{d x} [e^{x} y^{3}] + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.3.3

Move $2$ to the left of $\sin (y)$ .

$2 \sin (y) x + \frac{d}{d x} [e^{x} y^{3}] + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.4

Evaluate $\frac{d}{d x} [e^{x} y^{3}]$ .

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Step 8.4.1

Since $y^{3}$ is constant with respect to $x$ , the derivative of $e^{x} y^{3}$ with respect to $x$ is $y^{3} \frac{d}{d x} [e^{x}]$ .

$2 \sin (y) x + y^{3} \frac{d}{d x} [e^{x}] + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.4.2

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

$2 \sin (y) x + y^{3} e^{x} + \frac{d}{d x} [g (x)] = 2 x \sin (y) + y^{3} e^{x}$

Step 8.5

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

$2 \sin (y) x + y^{3} e^{x} + \frac{d g}{d x} = 2 x \sin (y) + y^{3} e^{x}$

Step 8.6

Simplify.

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Step 8.6.1

Reorder terms.

$\frac{d g}{d x} + 2 x \sin (y) + e^{x} y^{3} = 2 x \sin (y) + y^{3} e^{x}$

Step 8.6.2

Reorder factors in $\frac{d g}{d x} + 2 x \sin (y) + e^{x} y^{3}$ .

$\frac{d g}{d x} + 2 x \sin (y) + y^{3} e^{x} = 2 x \sin (y) + y^{3} e^{x}$

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 $2 x \sin (y)$ from both sides of the equation.

$\frac{d g}{d x} + y^{3} e^{x} = 2 x \sin (y) + y^{3} e^{x} - 2 x \sin (y)$

Step 9.1.2

Subtract $y^{3} e^{x}$ from both sides of the equation.

$\frac{d g}{d x} = 2 x \sin (y) + y^{3} e^{x} - 2 x \sin (y) - y^{3} e^{x}$

Step 9.1.3

Combine the opposite terms in $2 x \sin (y) + y^{3} e^{x} - 2 x \sin (y) - y^{3} e^{x}$ .

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Step 9.1.3.1

Subtract $2 x \sin (y)$ from $2 x \sin (y)$ .

$\frac{d g}{d x} = y^{3} e^{x} + 0 - y^{3} e^{x}$

Step 9.1.3.2

Add $y^{3} e^{x}$ and $0$ .

$\frac{d g}{d x} = y^{3} e^{x} - y^{3} e^{x}$

Step 9.1.3.3

Subtract $y^{3} e^{x}$ from $y^{3} e^{x}$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

$g (x) = 0 + C$

Step 10.4

Add $0$ and $C$ .

$g (x) = C$

Step 11

Substitute for $g (x)$ in $f (x, y) = x^{2} \sin (y) + e^{x} y^{3} + g (x)$ .

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

Step 12

Reorder factors in $f (x, y) = x^{2} \sin (y) + e^{x} y^{3} + C$ .

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