Calculus Examples

$y (x^{4} - y^{2}) d x + x (x^{4} + y^{2}) d y = 0$

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = x^{4} - y^{2}$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = y (0 + \frac{d}{d y} [- y^{2}]) + (x^{4} - y^{2}) \frac{d}{d y} [y]$

Step 1.3.3

Add $0$ and $\frac{d}{d y} [- y^{2}]$ .

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

Step 1.3.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- y^{2}$ with respect to $y$ is $- \frac{d}{d y} [y^{2}]$ .

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

Step 1.3.5

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

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

Step 1.3.6

Multiply $2$ by $- 1$ .

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

Step 1.4

Raise $y$ to the power of $1$ .

$\frac{\partial M}{\partial y} = - 2 (y^{1} y) + (x^{4} - y^{2}) \frac{d}{d y} [y]$

Step 1.5

Raise $y$ to the power of $1$ .

$\frac{\partial M}{\partial y} = - 2 (y^{1} y^{1}) + (x^{4} - y^{2}) \frac{d}{d y} [y]$

Step 1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\partial M}{\partial y} = - 2 y^{1 + 1} + (x^{4} - y^{2}) \frac{d}{d y} [y]$

Step 1.7

Add $1$ and $1$ .

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

Step 1.8

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

$\frac{\partial M}{\partial y} = - 2 y^{2} + (x^{4} - y^{2}) \cdot 1$

Step 1.9

Simplify by adding terms.

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Step 1.9.1

Multiply $x^{4} - y^{2}$ by $1$ .

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

Step 1.9.2

Subtract $y^{2}$ from $- 2 y^{2}$ .

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

Step 2

Find $\frac{\partial N}{\partial x}$ where $N (x, y) = x (x^{4} + y^{2})$ .

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{4} + y^{2}$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

$\frac{\partial N}{\partial x} = x (4 x^{3} + 0) + (x^{4} + y^{2}) \frac{d}{d x} [x]$

Step 2.3.4

Add $4 x^{3}$ and $0$ .

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

Step 2.4

Raise $x$ to the power of $1$ .

$\frac{\partial N}{\partial x} = 4 (x^{1} x^{3}) + (x^{4} + y^{2}) \frac{d}{d x} [x]$

Step 2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\partial N}{\partial x} = 4 x^{1 + 3} + (x^{4} + y^{2}) \frac{d}{d x} [x]$

Step 2.6

Add $1$ and $3$ .

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

Step 2.7

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

$\frac{\partial N}{\partial x} = 4 x^{4} + (x^{4} + y^{2}) \cdot 1$

Step 2.8

Simplify by adding terms.

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Step 2.8.1

Multiply $x^{4} + y^{2}$ by $1$ .

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

Step 2.8.2

Add $4 x^{4}$ and $x^{4}$ .

$\frac{\partial N}{\partial x} = 5 x^{4} + y^{2}$

Step 3

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

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Step 3.1

Substitute $x^{4} - 3 y^{2}$ for $\frac{\partial M}{\partial y}$ and $5 x^{4} + y^{2}$ for $\frac{\partial N}{\partial x}$ .

$x^{4} - 3 y^{2} = 5 x^{4} + y^{2}$

Step 3.2

Since the left side does not equal the right side, the equation is not an identity.

$x^{4} - 3 y^{2} = 5 x^{4} + y^{2}$ is not an identity.

Step 4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

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Step 4.1

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

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

Step 4.2

Substitute $5 x^{4} + y^{2}$ for $\frac{\partial N}{\partial x}$ .

$\frac{x^{4} - 3 y^{2} - (5 x^{4} + y^{2})}{N}$

Step 4.3

Substitute $x (x^{4} + y^{2})$ for $N$ .

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Step 4.3.1

Substitute $x (x^{4} + y^{2})$ for $N$ .

$\frac{x^{4} - 3 y^{2} - (5 x^{4} + y^{2})}{x (x^{4} + y^{2})}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Let $u = y^{2}$ . Substitute $u$ for all occurrences of $y^{2}$ .

$\frac{- 4 x^{4} - 4 u}{x (x^{4} + y^{2})}$

Step 4.3.2.2

Factor $4$ out of $- 4 x^{4} - 4 u$ .

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Step 4.3.2.2.1

Factor $4$ out of $- 4 x^{4}$ .

$\frac{4 (- x^{4}) - 4 u}{x (x^{4} + y^{2})}$

Step 4.3.2.2.2

Factor $4$ out of $- 4 u$ .

$\frac{4 (- x^{4}) + 4 (- u)}{x (x^{4} + y^{2})}$

Step 4.3.2.2.3

Factor $4$ out of $4 (- x^{4}) + 4 (- u)$ .

$\frac{4 (- x^{4} - u)}{x (x^{4} + y^{2})}$

Step 4.3.2.3

Replace all occurrences of $u$ with $y^{2}$ .

$\frac{4 (- x^{4} - y^{2})}{x (x^{4} + y^{2})}$

Step 4.3.3

Cancel the common factor of $- x^{4} - y^{2}$ and $x^{4} + y^{2}$ .

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Step 4.3.3.1

Factor $- 1$ out of $- x^{4}$ .

$\frac{4 (- (x^{4}) - y^{2})}{x (x^{4} + y^{2})}$

Step 4.3.3.2

Factor $- 1$ out of $- y^{2}$ .

$\frac{4 (- (x^{4}) - (y^{2}))}{x (x^{4} + y^{2})}$

Step 4.3.3.3

Factor $- 1$ out of $- (x^{4}) - (y^{2})$ .

$\frac{4 (- (x^{4} + y^{2}))}{x (x^{4} + y^{2})}$

Step 4.3.3.4

Rewrite $- (x^{4} + y^{2})$ as $- 1 (x^{4} + y^{2})$ .

$\frac{4 (- 1 (x^{4} + y^{2}))}{x (x^{4} + y^{2})}$

Step 4.3.3.5

Cancel the common factor.

$\frac{4 (- 1 (x^{4} + y^{2}))}{x (x^{4} + y^{2})}$

Step 4.3.3.6

Rewrite the expression.

$\frac{4 \cdot (- 1)}{x}$

Step 4.3.4

Multiply $4$ by $- 1$ .

$\frac{- 4}{x}$

Step 4.3.5

Move the negative in front of the fraction.

$- \frac{4}{x}$

Step 4.4

Find the integration factor $μ (x, y) = e^{\int \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} d x}$ .

$μ (x, y) = e^{\int - \frac{4}{x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{4}{x} d x}$ .

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Step 5.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$μ (x, y) = e^{- \int \frac{4}{x} d x}$

Step 5.2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$μ (x, y) = e^{- (4 \int \frac{1}{x} d x)}$

Step 5.3

Multiply $4$ by $- 1$ .

$μ (x, y) = e^{- 4 \int \frac{1}{x} d x}$

Step 5.4

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$μ (x, y) = e^{- 4 (\ln (| x |) + C)}$

Step 5.5

Simplify.

$μ (x, y) = e^{- 4 \ln (| x |)} + C$

Step 5.6

Simplify each term.

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Step 5.6.1

Simplify $- 4 \ln (| x |)$ by moving $- 4$ inside the logarithm.

$μ (x, y) = e^{\ln ({| x |}^{- 4})} + C$

Step 5.6.2

Exponentiation and log are inverse functions.

$μ (x, y) = {| x |}^{- 4} + C$

Step 5.6.3

Remove the absolute value in ${| x |}^{- 4}$ because exponentiations with even powers are always positive.

$μ (x, y) = x^{- 4} + C$

Step 5.6.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$μ (x, y) = \frac{1}{x^{4}} + C$

Step 6

Multiply both sides of $y (x^{4} - y^{2}) d x + x (x^{4} + y^{2}) d y = 0$ by the integration factor $\frac{1}{x^{4}}$ .

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Step 6.1

Multiply $y (x^{4} - y^{2})$ by $\frac{1}{x^{4}}$ .

$y (x^{4} - y^{2}) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.2

Apply the distributive property.

$(y x^{4} + y (- y^{2})) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.3

Rewrite using the commutative property of multiplication.

$(y x^{4} - y \cdot y^{2}) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.4

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 6.4.1

Move $y^{2}$ .

$(y x^{4} - (y^{2} y)) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.4.2

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

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Step 6.4.2.1

Raise $y$ to the power of $1$ .

$(y x^{4} - (y^{2} y^{1})) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(y x^{4} - y^{2 + 1}) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.4.3

Add $2$ and $1$ .

$(y x^{4} - y^{3}) \frac{1}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.5

Multiply $y x^{4} - y^{3}$ by $\frac{1}{x^{4}}$ .

$\frac{y x^{4} - y^{3}}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Factor $y$ out of $y x^{4} - y^{3}$ .

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Step 6.6.1.1

Factor $y$ out of $y x^{4}$ .

$\frac{y (x^{4}) - y^{3}}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.6.1.2

Factor $y$ out of $- y^{3}$ .

$\frac{y (x^{4}) + y (- y^{2})}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.6.1.3

Factor $y$ out of $y (x^{4}) + y (- y^{2})$ .

$\frac{y (x^{4} - y^{2})}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.6.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{y ({(x^{2})}^{2} - y^{2})}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = y$ .

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + x (x^{4} + y^{2}) d y = 0$

Step 6.7

Multiply $x (x^{4} + y^{2})$ by $\frac{1}{x^{4}}$ .

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + x (x^{4} + y^{2}) \frac{1}{x^{4}} d y = 0$

Step 6.8

Cancel the common factor of $x$ .

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Step 6.8.1

Factor $x$ out of $x^{4}$ .

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + x (x^{4} + y^{2}) \frac{1}{x \cdot x^{3}} d y = 0$

Step 6.8.2

Cancel the common factor.

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + x (x^{4} + y^{2}) \frac{1}{x \cdot x^{3}} d y = 0$

Step 6.8.3

Rewrite the expression.

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + (x^{4} + y^{2}) \frac{1}{x^{3}} d y = 0$

Step 6.9

Multiply $x^{4} + y^{2}$ by $\frac{1}{x^{3}}$ .

$\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} d x + \frac{x^{4} + y^{2}}{x^{3}} d y = 0$

Step 7

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

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

Step 8

Integrate $N (x, y) = \frac{x^{4} + y^{2}}{x^{3}}$ to find $f (x, y)$ .

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Step 8.1

Since $\frac{1}{x^{3}}$ is constant with respect to $y$ , move $\frac{1}{x^{3}}$ out of the integral.

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

Step 8.2

Split the single integral into multiple integrals.

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

Step 8.3

Apply the constant rule.

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Simplify.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [\frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3}) + g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.2

By the Sum Rule, the derivative of $\frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3}) + g (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3})] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [\frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3})] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3

Evaluate $\frac{d}{d x} [\frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3})]$ .

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Step 11.3.1

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

$\frac{d}{d x} [\frac{1}{x^{3}} (x^{4} y + \frac{y^{3}}{3})] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \frac{1}{x^{3}}$ and $g (x) = x^{4} y + \frac{y^{3}}{3}$ .

$\frac{1}{x^{3}} \frac{d}{d x} [x^{4} y + \frac{y^{3}}{3}] + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.3

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

$\frac{1}{x^{3}} (\frac{d}{d x} [x^{4} y] + \frac{d}{d x} [\frac{y^{3}}{3}]) + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.4

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

$\frac{1}{x^{3}} (y \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{y^{3}}{3}]) + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.5

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

$\frac{1}{x^{3}} (y (4 x^{3}) + \frac{d}{d x} [\frac{y^{3}}{3}]) + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.6

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

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.7

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 11.3.8.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.8.2

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

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) (- u^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.8.3

Replace all occurrences of $u$ with $x^{3}$ .

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.9

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

$\frac{1}{x^{3}} (y (4 x^{3}) + 0) + (x^{4} y + \frac{y^{3}}{3}) (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.10

Move $4$ to the left of $y$ .

$\frac{1}{x^{3}} (4 \cdot y x^{3} + 0) + (x^{4} y + \frac{y^{3}}{3}) (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.11

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

$\frac{1}{x^{3}} (4 y x^{3}) + (x^{4} y + \frac{y^{3}}{3}) (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.12

Combine $4$ and $\frac{1}{x^{3}}$ .

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

Step 11.3.13

Combine $y$ and $\frac{4}{x^{3}}$ .

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

Step 11.3.14

Combine $\frac{y \cdot 4}{x^{3}}$ and $x^{3}$ .

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

Step 11.3.15

Move $4$ to the left of $y$ .

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

Step 11.3.16

Cancel the common factor of $x^{3}$ .

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Step 11.3.16.1

Cancel the common factor.

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

Step 11.3.16.2

Divide $4 y$ by $1$ .

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

Step 11.3.17

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 11.3.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 11.3.17.2

Multiply $3$ by $- 2$ .

$4 y + (x^{4} y + \frac{y^{3}}{3}) (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.18

Multiply $3$ by $- 1$ .

$4 y + (x^{4} y + \frac{y^{3}}{3}) (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.19

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 11.3.19.1

Move $x^{2}$ .

$4 y + (x^{4} y + \frac{y^{3}}{3}) (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.19.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 y + (x^{4} y + \frac{y^{3}}{3}) (- 3 x^{2 - 6}) + \frac{d}{d x} [g (x)] = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.3.19.3

Subtract $6$ from $2$ .

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 y + (x^{4} y + \frac{y^{3}}{3}) (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.2

Apply the distributive property.

$4 y + x^{4} y (- 3 \frac{1}{x^{4}}) + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3

Combine terms.

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Step 11.5.3.1

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$4 y + x^{4} y \frac{- 3}{x^{4}} + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.2

Move the negative in front of the fraction.

$4 y + x^{4} y (- \frac{3}{x^{4}}) + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.3

Combine $x^{4}$ and $\frac{3}{x^{4}}$ .

$4 y + y (- \frac{x^{4} \cdot 3}{x^{4}}) + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.4

Combine $y$ and $\frac{x^{4} \cdot 3}{x^{4}}$ .

$4 y - \frac{y (x^{4} \cdot 3)}{x^{4}} + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.5

Move $3$ to the left of $x^{4}$ .

$4 y - \frac{y (3 \cdot x^{4})}{x^{4}} + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.6

Move $3$ to the left of $y$ .

$4 y - \frac{3 \cdot y x^{4}}{x^{4}} + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.7

Cancel the common factor of $x^{4}$ .

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Step 11.5.3.7.1

Cancel the common factor.

$4 y - \frac{3 y x^{4}}{x^{4}} + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.7.2

Divide $3 y$ by $1$ .

$4 y - (3 y) + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.8

Multiply $3$ by $- 1$ .

$4 y - 3 y + \frac{y^{3}}{3} (- 3 \frac{1}{x^{4}}) + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.9

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$4 y - 3 y + \frac{y^{3}}{3} \cdot \frac{- 3}{x^{4}} + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.10

Move the negative in front of the fraction.

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

Step 11.5.3.11

Multiply $\frac{y^{3}}{3}$ by $\frac{3}{x^{4}}$ .

$4 y - 3 y - \frac{y^{3} \cdot 3}{3 x^{4}} + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.12

Move $3$ to the left of $y^{3}$ .

$4 y - 3 y - \frac{3 \cdot y^{3}}{3 x^{4}} + \frac{d g}{d x} = \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$

Step 11.5.3.13

Cancel the common factor of $3$ .

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Step 11.5.3.13.1

Cancel the common factor.

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

Step 11.5.3.13.2

Rewrite the expression.

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

Step 11.5.3.14

Subtract $3 y$ from $4 y$ .

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

Step 11.5.4

Reorder terms.

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

Step 12

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

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Step 12.1

Move all terms containing variables to the left side of the equation.

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Step 12.1.1

Subtract $\frac{y (x^{2} + y) (x^{2} - y)}{x^{4}}$ from both sides of the equation.

$\frac{d g}{d x} + y - \frac{y^{3}}{x^{4}} - \frac{y (x^{2} + y) (x^{2} - y)}{x^{4}} = 0$

Step 12.1.2

Combine the numerators over the common denominator.

$\frac{d g}{d x} + y + \frac{- y^{3} - y (x^{2} + y) (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3

Simplify each term.

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Step 12.1.3.1

Apply the distributive property.

$\frac{d g}{d x} + y + \frac{- y^{3} + (- y x^{2} - y \cdot y) (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3.2

Multiply $y$ by $y$ by adding the exponents.

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Step 12.1.3.2.1

Move $y$ .

$\frac{d g}{d x} + y + \frac{- y^{3} + (- y x^{2} - (y \cdot y)) (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3.2.2

Multiply $y$ by $y$ .

$\frac{d g}{d x} + y + \frac{- y^{3} + (- y x^{2} - y^{2}) (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3.3

Expand $(- y x^{2} - y^{2}) (x^{2} - y)$ using the FOIL Method.

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Step 12.1.3.3.1

Apply the distributive property.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{2} (x^{2} - y) - y^{2} (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3.3.2

Apply the distributive property.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{2} x^{2} - y x^{2} (- y) - y^{2} (x^{2} - y)}{x^{4}} = 0$

Step 12.1.3.3.3

Apply the distributive property.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{2} x^{2} - y x^{2} (- y) - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4

Simplify and combine like terms.

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Step 12.1.3.4.1

Simplify each term.

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Step 12.1.3.4.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 12.1.3.4.1.1.1

Move $x^{2}$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y (x^{2} x^{2}) - y x^{2} (- y) - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{2 + 2} - y x^{2} (- y) - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.1.3

Add $2$ and $2$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} - y x^{2} (- y) - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 12.1.3.4.1.2.1

Move $y$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} - (y \cdot y) x^{2} \cdot - 1 - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.2.2

Multiply $y$ by $y$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} - y^{2} x^{2} \cdot - 1 - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.3

Multiply $- y^{2} x^{2} \cdot - 1$ .

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Step 12.1.3.4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + 1 y^{2} x^{2} - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.3.2

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

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - y^{2} (- y)}{x^{4}} = 0$

Step 12.1.3.4.1.4

Rewrite using the commutative property of multiplication.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{2} y}{x^{4}} = 0$

Step 12.1.3.4.1.5

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 12.1.3.4.1.5.1

Move $y$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - 1 \cdot - 1 (y \cdot y^{2})}{x^{4}} = 0$

Step 12.1.3.4.1.5.2

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

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Step 12.1.3.4.1.5.2.1

Raise $y$ to the power of $1$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - 1 \cdot - 1 (y^{1} y^{2})}{x^{4}} = 0$

Step 12.1.3.4.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{1 + 2}}{x^{4}} = 0$

Step 12.1.3.4.1.5.3

Add $1$ and $2$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} - 1 \cdot - 1 y^{3}}{x^{4}} = 0$

Step 12.1.3.4.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} + 1 y^{3}}{x^{4}} = 0$

Step 12.1.3.4.1.7

Multiply $y^{3}$ by $1$ .

$\frac{d g}{d x} + y + \frac{- y^{3} - y x^{4} + y^{2} x^{2} - y^{2} x^{2} + y^{3}}{x^{4}} = 0$

Step 12.1.3.4.2

Subtract $y^{2} x^{2}$ from $y^{2} x^{2}$ .

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

Step 12.1.3.4.3

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

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

Step 12.1.4

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

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Step 12.1.4.1

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

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

Step 12.1.4.2

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

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

Step 12.1.5

Cancel the common factor of $x^{4}$ .

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Step 12.1.5.1

Cancel the common factor.

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

Step 12.1.5.2

Divide $- y$ by $1$ .

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

Step 12.1.6

Combine the opposite terms in $\frac{d g}{d x} + y - y$ .

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Step 12.1.6.1

Subtract $y$ from $y$ .

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

Step 12.1.6.2

Add $\frac{d g}{d x}$ and $0$ .

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

Step 13

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

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Step 13.1

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

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

Step 13.2

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

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

Step 13.3

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

$g (x) = 0 + C$

Step 13.4

Add $0$ and $C$ .

$g (x) = C$

Step 14

Substitute for $g (x)$ in $f (x, y) = \frac{1}{x^{3}} (x^{4} y + \frac{1}{3} y^{3}) + g (x)$ .

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

Step 15

Simplify each term.

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Step 15.1

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

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

Step 15.2

Multiply $\frac{1}{x^{3}}$ by $x^{4} y + \frac{y^{3}}{3}$ .

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

Step 15.3

Simplify the numerator.

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Step 15.3.1

Factor $y$ out of $x^{4} y + \frac{y^{3}}{3}$ .

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Step 15.3.1.1

Factor $y$ out of $x^{4} y$ .

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

Step 15.3.1.2

Factor $y$ out of $\frac{y^{3}}{3}$ .

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

Step 15.3.1.3

Factor $y$ out of $y x^{4} + y \frac{y^{2}}{3}$ .

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

Step 15.3.2

To write $x^{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 15.3.3

Combine $x^{4}$ and $\frac{3}{3}$ .

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

Step 15.3.4

Combine the numerators over the common denominator.

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

Step 15.3.5

Move $3$ to the left of $x^{4}$ .

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

Step 15.4

Combine $y$ and $\frac{3 x^{4} + y^{2}}{3}$ .

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

Step 15.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 15.6

Combine.

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

Step 15.7

Multiply $y$ by $1$ .

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