Calculus Examples

$(2 x + x^{3} + 3 y) d x + (3 x + 2 y) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [x^{3}] + \frac{d}{d y} [3 y]$

Step 1.2.3

Since $x^{3}$ is constant with respect to $y$ , the derivative of $x^{3}$ with respect to $y$ is $0$ .

$\frac{\partial M}{\partial y} = 0 + 0 + \frac{d}{d y} [3 y]$

Step 1.3

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

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Step 1.3.1

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

$\frac{\partial M}{\partial y} = 0 + 0 + 3 \frac{d}{d y} [y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 + 0 + 3 \cdot 1$

Step 1.3.3

Multiply $3$ by $1$ .

$\frac{\partial M}{\partial y} = 0 + 0 + 3$

Step 1.4

Combine terms.

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Step 1.4.1

Add $0$ and $0$ .

$\frac{\partial M}{\partial y} = 0 + 3$

Step 1.4.2

Add $0$ and $3$ .

$\frac{\partial M}{\partial y} = 3$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [3 x + 2 y]$

Step 2.2

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

$\frac{\partial N}{\partial x} = \frac{d}{d x} [3 x] + \frac{d}{d x} [2 y]$

Step 2.3

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

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Step 2.3.1

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

$\frac{\partial N}{\partial x} = 3 \frac{d}{d x} [x] + \frac{d}{d x} [2 y]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 3 \cdot 1 + \frac{d}{d x} [2 y]$

Step 2.3.3

Multiply $3$ by $1$ .

$\frac{\partial N}{\partial x} = 3 + \frac{d}{d x} [2 y]$

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 $0$ .

$\frac{\partial N}{\partial x} = 3 + 0$

Step 2.4.2

Add $3$ and $0$ .

$\frac{\partial N}{\partial x} = 3$

Step 3

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

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Step 3.1

Substitute $3$ for $\frac{\partial M}{\partial y}$ and $3$ for $\frac{\partial N}{\partial x}$ .

$3 = 3$

Step 3.2

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

$3 = 3$ is an identity.

Step 4

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

$f (x, y) = \int 3 x + 2 y d y$

Step 5

Integrate $N (x, y) = 3 x + 2 y$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int 3 x d y + \int 2 y d y$

Step 5.2

Apply the constant rule.

$f (x, y) = 3 x y + C + \int 2 y d y$

Step 5.3

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

$f (x, y) = 3 x y + C + 2 \int y d y$

Step 5.4

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

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

Step 5.5

Simplify.

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

Step 5.6

Simplify.

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Step 5.6.1

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

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

Step 5.6.2

Cancel the common factor of $2$ .

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Step 5.6.2.1

Cancel the common factor.

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

Step 5.6.2.2

Rewrite the expression.

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

Step 5.6.3

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

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

Step 6

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

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

Step 7

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

$\frac{\partial f}{\partial x} = 2 x + x^{3} + 3 y$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $x$ .

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

$3 y \cdot 1 + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [g (x)] = 2 x + x^{3} + 3 y$

Step 8.3.3

Multiply $3$ by $1$ .

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

Step 8.4

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

$3 y + 0 + \frac{d}{d x} [g (x)] = 2 x + x^{3} + 3 y$

Step 8.5

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

$3 y + 0 + \frac{d g}{d x} = 2 x + x^{3} + 3 y$

Step 8.6

Simplify.

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Step 8.6.1

Add $3 y$ and $0$ .

$3 y + \frac{d g}{d x} = 2 x + x^{3} + 3 y$

Step 8.6.2

Reorder terms.

$\frac{d g}{d x} + 3 y = 2 x + x^{3} + 3 y$

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 $3 y$ from both sides of the equation.

$\frac{d g}{d x} = 2 x + x^{3} + 3 y - 3 y$

Step 9.1.2

Combine the opposite terms in $2 x + x^{3} + 3 y - 3 y$ .

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Step 9.1.2.1

Subtract $3 y$ from $3 y$ .

$\frac{d g}{d x} = 2 x + x^{3} + 0$

Step 9.1.2.2

Add $2 x + x^{3}$ and $0$ .

$\frac{d g}{d x} = 2 x + x^{3}$

Step 10

Find the antiderivative of $2 x + x^{3}$ to find $g (x)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d x} = 2 x + x^{3}$ .

$\int \frac{d g}{d x} d x = \int 2 x + x^{3} d x$

Step 10.2

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

$g (x) = \int 2 x + x^{3} d x$

Step 10.3

Split the single integral into multiple integrals.

$g (x) = \int 2 x d x + \int x^{3} d x$

Step 10.4

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

$g (x) = 2 \int x d x + \int x^{3} d x$

Step 10.5

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

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

Step 10.6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$g (x) = 2 (\frac{1}{2} x^{2} + C) + \frac{1}{4} x^{4} + C$

Step 10.7

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

$g (x) = 2 (\frac{x^{2}}{2} + C) + \frac{1}{4} x^{4} + C$

Step 10.8

Simplify.

$g (x) = x^{2} + \frac{1}{4} x^{4} + C$

Step 11

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

$f (x, y) = 3 x y + y^{2} + x^{2} + \frac{1}{4} x^{4} + C$

Step 12

Combine $\frac{1}{4}$ and $x^{4}$ .

$f (x, y) = 3 x y + y^{2} + x^{2} + \frac{x^{4}}{4} + C$