Calculus Examples

$(15 x^{2} y^{2} - y^{4}) d x + (10 x^{3} y - 4 x y^{3} - 5 y^{4}) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 15 x^{2} y^{2} - y^{4}$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [15 x^{2} y^{2} - y^{4}]$

Step 1.2

By the Sum Rule, the derivative of $15 x^{2} y^{2} - y^{4}$ with respect to $y$ is $\frac{d}{d y} [15 x^{2} y^{2}] + \frac{d}{d y} [- y^{4}]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [15 x^{2} y^{2}] + \frac{d}{d y} [- y^{4}]$

Step 1.3

Evaluate $\frac{d}{d y} [15 x^{2} y^{2}]$ .

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Step 1.3.1

Since $15 x^{2}$ is constant with respect to $y$ , the derivative of $15 x^{2} y^{2}$ with respect to $y$ is $15 x^{2} \frac{d}{d y} [y^{2}]$ .

$\frac{\partial M}{\partial y} = 15 x^{2} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- y^{4}]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 15 x^{2} (2 y) + \frac{d}{d y} [- y^{4}]$

Step 1.3.3

Multiply $2$ by $15$ .

$\frac{\partial M}{\partial y} = 30 x^{2} y + \frac{d}{d y} [- y^{4}]$

Step 1.4

Evaluate $\frac{d}{d y} [- y^{4}]$ .

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Step 1.4.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{4}$ with respect to $y$ is $- \frac{d}{d y} [y^{4}]$ .

$\frac{\partial M}{\partial y} = 30 x^{2} y - \frac{d}{d y} [y^{4}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$\frac{\partial M}{\partial y} = 30 x^{2} y - (4 y^{3})$

Step 1.4.3

Multiply $4$ by $- 1$ .

$\frac{\partial M}{\partial y} = 30 x^{2} y - 4 y^{3}$

Step 1.5

Reorder terms.

$\frac{\partial M}{\partial y} = - 4 y^{3} + 30 x^{2} y$

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [10 x^{3} y - 4 x y^{3} - 5 y^{4}]$

Step 2.2

By the Sum Rule, the derivative of $10 x^{3} y - 4 x y^{3} - 5 y^{4}$ with respect to $x$ is $\frac{d}{d x} [10 x^{3} y] + \frac{d}{d x} [- 4 x y^{3}] + \frac{d}{d x} [- 5 y^{4}]$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [10 x^{3} y] + \frac{d}{d x} [- 4 x y^{3}] + \frac{d}{d x} [- 5 y^{4}]$

Step 2.3

Evaluate $\frac{d}{d x} [10 x^{3} y]$ .

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Step 2.3.1

Since $10 y$ is constant with respect to $x$ , the derivative of $10 x^{3} y$ with respect to $x$ is $10 y \frac{d}{d x} [x^{3}]$ .

$\frac{\partial N}{\partial x} = 10 y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x y^{3}] + \frac{d}{d x} [- 5 y^{4}]$

Step 2.3.2

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

$\frac{\partial N}{\partial x} = 10 y (3 x^{2}) + \frac{d}{d x} [- 4 x y^{3}] + \frac{d}{d x} [- 5 y^{4}]$

Step 2.3.3

Multiply $3$ by $10$ .

$\frac{\partial N}{\partial x} = 30 y x^{2} + \frac{d}{d x} [- 4 x y^{3}] + \frac{d}{d x} [- 5 y^{4}]$

Step 2.4

Evaluate $\frac{d}{d x} [- 4 x y^{3}]$ .

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Step 2.4.1

Since $- 4 y^{3}$ is constant with respect to $x$ , the derivative of $- 4 x y^{3}$ with respect to $x$ is $- 4 y^{3} \frac{d}{d x} [x]$ .

$\frac{\partial N}{\partial x} = 30 y x^{2} - 4 y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [- 5 y^{4}]$

Step 2.4.2

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

$\frac{\partial N}{\partial x} = 30 y x^{2} - 4 y^{3} \cdot 1 + \frac{d}{d x} [- 5 y^{4}]$

Step 2.4.3

Multiply $- 4$ by $1$ .

$\frac{\partial N}{\partial x} = 30 y x^{2} - 4 y^{3} + \frac{d}{d x} [- 5 y^{4}]$

Step 2.5

Since $- 5 y^{4}$ is constant with respect to $x$ , the derivative of $- 5 y^{4}$ with respect to $x$ is $0$ .

$\frac{\partial N}{\partial x} = 30 y x^{2} - 4 y^{3} + 0$

Step 2.6

Simplify.

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Step 2.6.1

Add $30 y x^{2} - 4 y^{3}$ and $0$ .

$\frac{\partial N}{\partial x} = 30 y x^{2} - 4 y^{3}$

Step 2.6.2

Reorder terms.

$\frac{\partial N}{\partial x} = - 4 y^{3} + 30 x^{2} y$

Step 3

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

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Step 3.1

Substitute $- 4 y^{3} + 30 x^{2} y$ for $\frac{\partial M}{\partial y}$ and $- 4 y^{3} + 30 x^{2} y$ for $\frac{\partial N}{\partial x}$ .

$- 4 y^{3} + 30 x^{2} y = - 4 y^{3} + 30 x^{2} y$

Step 3.2

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

$- 4 y^{3} + 30 x^{2} y = - 4 y^{3} + 30 x^{2} y$ is an identity.

Step 4

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

$f (x, y) = \int 15 x^{2} y^{2} - y^{4} d x$

Step 5

Integrate $M (x, y) = 15 x^{2} y^{2} - y^{4}$ to find $f (x, y)$ .

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Step 5.1

Split the single integral into multiple integrals.

$f (x, y) = \int 15 x^{2} y^{2} d x + \int - y^{4} d x$

Step 5.2

Since $15 y^{2}$ is constant with respect to $x$ , move $15 y^{2}$ out of the integral.

$f (x, y) = 15 y^{2} \int x^{2} d x + \int - y^{4} d x$

Step 5.3

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

$f (x, y) = 15 y^{2} (\frac{1}{3} x^{3} + C) + \int - y^{4} d x$

Step 5.4

Apply the constant rule.

$f (x, y) = 15 y^{2} (\frac{1}{3} x^{3} + C) - y^{4} x + C$

Step 5.5

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

$f (x, y) = 15 y^{2} (\frac{x^{3}}{3} + C) - y^{4} x + C$

Step 5.6

Simplify.

$f (x, y) = 5 y^{2} x^{3} - y^{4} x + C$

Step 6

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

$f (x, y) = 5 y^{2} x^{3} - y^{4} x + g (y)$

Step 7

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

$\frac{\partial f}{\partial y} = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8

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

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Step 8.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [5 y^{2} x^{3} - y^{4} x + g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.2

By the Sum Rule, the derivative of $5 y^{2} x^{3} - y^{4} x + g (y)$ with respect to $y$ is $\frac{d}{d y} [5 y^{2} x^{3}] + \frac{d}{d y} [- y^{4} x] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [5 y^{2} x^{3}] + \frac{d}{d y} [- y^{4} x] + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.3

Evaluate $\frac{d}{d y} [5 y^{2} x^{3}]$ .

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Step 8.3.1

Since $5 x^{3}$ is constant with respect to $y$ , the derivative of $5 y^{2} x^{3}$ with respect to $y$ is $5 x^{3} \frac{d}{d y} [y^{2}]$ .

$5 x^{3} \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- y^{4} x] + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.3.2

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

$5 x^{3} (2 y) + \frac{d}{d y} [- y^{4} x] + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.3.3

Multiply $2$ by $5$ .

$10 x^{3} y + \frac{d}{d y} [- y^{4} x] + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.4

Evaluate $\frac{d}{d y} [- y^{4} x]$ .

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Step 8.4.1

Since $- x$ is constant with respect to $y$ , the derivative of $- y^{4} x$ with respect to $y$ is $- x \frac{d}{d y} [y^{4}]$ .

$10 x^{3} y - x \frac{d}{d y} [y^{4}] + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 4$ .

$10 x^{3} y - x (4 y^{3}) + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.4.3

Multiply $4$ by $- 1$ .

$10 x^{3} y - 4 x y^{3} + \frac{d}{d y} [g (y)] = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.5

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

$10 x^{3} y - 4 x y^{3} + \frac{d g}{d y} = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

Step 8.6

Reorder terms.

$\frac{d g}{d y} + 10 x^{3} y - 4 y^{3} x = 10 x^{3} y - 4 x y^{3} - 5 y^{4}$

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 $10 x^{3} y$ from both sides of the equation.

$\frac{d g}{d y} - 4 y^{3} x = 10 x^{3} y - 4 x y^{3} - 5 y^{4} - 10 x^{3} y$

Step 9.1.2

Add $4 y^{3} x$ to both sides of the equation.

$\frac{d g}{d y} = 10 x^{3} y - 4 x y^{3} - 5 y^{4} - 10 x^{3} y + 4 y^{3} x$

Step 9.1.3

Combine the opposite terms in $10 x^{3} y - 4 x y^{3} - 5 y^{4} - 10 x^{3} y + 4 y^{3} x$ .

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Step 9.1.3.1

Subtract $10 x^{3} y$ from $10 x^{3} y$ .

$\frac{d g}{d y} = - 4 x y^{3} - 5 y^{4} + 0 + 4 y^{3} x$

Step 9.1.3.2

Add $- 4 x y^{3} - 5 y^{4}$ and $0$ .

$\frac{d g}{d y} = - 4 x y^{3} - 5 y^{4} + 4 y^{3} x$

Step 9.1.3.3

Reorder the factors in the terms $- 4 x y^{3}$ and $4 y^{3} x$ .

$\frac{d g}{d y} = - 4 y^{3} x - 5 y^{4} + 4 y^{3} x$

Step 9.1.3.4

Add $- 4 y^{3} x$ and $4 y^{3} x$ .

$\frac{d g}{d y} = - 5 y^{4} + 0$

Step 9.1.3.5

Add $- 5 y^{4}$ and $0$ .

$\frac{d g}{d y} = - 5 y^{4}$

Step 10

Find the antiderivative of $- 5 y^{4}$ to find $g (y)$ .

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Step 10.1

Integrate both sides of $\frac{d g}{d y} = - 5 y^{4}$ .

$\int \frac{d g}{d y} d y = \int - 5 y^{4} d y$

Step 10.2

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

$g (y) = \int - 5 y^{4} d y$

Step 10.3

Since $- 5$ is constant with respect to $y$ , move $- 5$ out of the integral.

$g (y) = - 5 \int y^{4} d y$

Step 10.4

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$g (y) = - 5 (\frac{1}{5} y^{5} + C)$

Step 10.5

Simplify the answer.

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Step 10.5.1

Rewrite $- 5 (\frac{1}{5} y^{5} + C)$ as $- 5 (\frac{1}{5}) y^{5} + C$ .

$g (y) = - 5 (\frac{1}{5}) y^{5} + C$

Step 10.5.2

Simplify.

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Step 10.5.2.1

Combine $- 5$ and $\frac{1}{5}$ .

$g (y) = \frac{- 5}{5} y^{5} + C$

Step 10.5.2.2

Cancel the common factor of $- 5$ and $5$ .

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Step 10.5.2.2.1

Factor $5$ out of $- 5$ .

$g (y) = \frac{5 \cdot - 1}{5} y^{5} + C$

Step 10.5.2.2.2

Cancel the common factors.

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Step 10.5.2.2.2.1

Factor $5$ out of $5$ .

$g (y) = \frac{5 \cdot - 1}{5 (1)} y^{5} + C$

Step 10.5.2.2.2.2

Cancel the common factor.

$g (y) = \frac{5 \cdot - 1}{5 \cdot 1} y^{5} + C$

Step 10.5.2.2.2.3

Rewrite the expression.

$g (y) = \frac{- 1}{1} y^{5} + C$

Step 10.5.2.2.2.4

Divide $- 1$ by $1$ .

$g (y) = - y^{5} + C$

Step 11

Substitute for $g (y)$ in $f (x, y) = 5 y^{2} x^{3} - y^{4} x + g (y)$ .

$f (x, y) = 5 y^{2} x^{3} - y^{4} x - y^{5} + C$