Calculus Examples

$5 x^{2} (y d) x + (x^{3} + y^{3}) d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = 5 x^{2} y$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [5 x^{2} y]$

Step 1.2

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

$\frac{\partial M}{\partial y} = 5 x^{2} \frac{d}{d y} [y]$

Step 1.3

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

$\frac{\partial M}{\partial y} = 5 x^{2} \cdot 1$

Step 1.4

Multiply $5$ by $1$ .

$\frac{\partial M}{\partial y} = 5 x^{2}$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$\frac{\partial N}{\partial x} = 3 x^{2} + 0$

Step 2.5

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

$\frac{\partial N}{\partial x} = 3 x^{2}$

Step 3

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

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Step 3.1

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

$5 x^{2} = 3 x^{2}$

Step 3.2

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

$5 x^{2} = 3 x^{2}$ is not an identity.

Step 4

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

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Step 4.1

Substitute $3 x^{2}$ for $\frac{\partial N}{\partial x}$ .

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

Step 4.2

Substitute $5 x^{2}$ for $\frac{\partial M}{\partial y}$ .

$\frac{3 x^{2} - (5 x^{2})}{M}$

Step 4.3

Substitute $5 x^{2} y$ for $M$ .

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Step 4.3.1

Substitute $5 x^{2} y$ for $M$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $x^{2}$ out of $3 x^{2} - 1 \cdot 5 x^{2}$ .

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Step 4.3.2.1.1

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} \cdot 3 - 1 \cdot 5 x^{2}}{5 x^{2} y}$

Step 4.3.2.1.2

Factor $x^{2}$ out of $- 1 \cdot 5 x^{2}$ .

$\frac{x^{2} \cdot 3 + x^{2} (- 1 \cdot 5)}{5 x^{2} y}$

Step 4.3.2.1.3

Factor $x^{2}$ out of $x^{2} \cdot 3 + x^{2} (- 1 \cdot 5)$ .

$\frac{x^{2} (3 - 1 \cdot 5)}{5 x^{2} y}$

Step 4.3.2.2

Multiply $- 1$ by $5$ .

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

Step 4.3.2.3

Subtract $5$ from $3$ .

$\frac{x^{2} \cdot - 2}{5 x^{2} y}$

Step 4.3.3

Cancel the common factor of $x^{2}$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{x^{2} \cdot - 2}{5 x^{2} y}$

Step 4.3.3.2

Rewrite the expression.

$\frac{- 2}{5 y}$

Step 4.3.4

Substitute $5 x^{2} y$ for $M$ .

$- \frac{2}{5 y}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{2}{5 y} d y}$

Step 5

Evaluate the integral $e^{\int - \frac{2}{5 y} d y}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{2}{5 y} d y}$

Step 5.2

Since $\frac{2}{5}$ is constant with respect to $y$ , move $\frac{2}{5}$ out of the integral.

$μ (x, y) = e^{- (\frac{2}{5} \int \frac{1}{y} d y)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{2}{5} (\ln (| y |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{2}{5} \ln (| y |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{2}{5} \ln (| y |)$ .

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Step 5.5.1.1

Reorder $\ln (| y |)$ and $\frac{2}{5}$ .

$μ (x, y) = e^{- (\frac{2}{5} \ln (| y |))} + C$

Step 5.5.1.2

Simplify $\frac{2}{5} \ln (| y |)$ by moving $\frac{2}{5}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| y |}^{\frac{2}{5}})} + C$

Step 5.5.2

Simplify $- \ln ({| y |}^{\frac{2}{5}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| y |}^{\frac{2}{5}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

$μ (x, y) = {({| y |}^{\frac{2}{5}})}^{- 1} + C$

Step 5.5.4

Multiply the exponents in ${({| y |}^{\frac{2}{5}})}^{- 1}$ .

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Step 5.5.4.1

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

$μ (x, y) = {| y |}^{\frac{2}{5} \cdot - 1} + C$

Step 5.5.4.2

Multiply $\frac{2}{5} \cdot - 1$ .

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Step 5.5.4.2.1

Combine $\frac{2}{5}$ and $- 1$ .

$μ (x, y) = {| y |}^{\frac{2 \cdot - 1}{5}} + C$

Step 5.5.4.2.2

Multiply $2$ by $- 1$ .

$μ (x, y) = {| y |}^{\frac{- 2}{5}} + C$

Step 5.5.4.3

Move the negative in front of the fraction.

$μ (x, y) = {| y |}^{- \frac{2}{5}} + C$

Step 5.5.5

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

$μ (x, y) = \frac{1}{{| y |}^{\frac{2}{5}}} + C$

Step 6

Multiply both sides of $5 x^{2} y d x + (x^{3} + y^{3}) d y = 0$ by the integration factor $\frac{1}{y^{\frac{2}{5}}}$ .

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Step 6.1

Multiply $5 x^{2} y$ by $\frac{1}{y^{\frac{2}{5}}}$ .

$5 x^{2} y \frac{1}{y^{\frac{2}{5}}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.2

Multiply $5 x^{2} y \frac{1}{y^{\frac{2}{5}}}$ .

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Step 6.2.1

Combine $5$ and $\frac{1}{y^{\frac{2}{5}}}$ .

$x^{2} y \frac{5}{y^{\frac{2}{5}}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.2.2

Combine $x^{2}$ and $\frac{5}{y^{\frac{2}{5}}}$ .

$y \frac{x^{2} \cdot 5}{y^{\frac{2}{5}}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.2.3

Combine $y$ and $\frac{x^{2} \cdot 5}{y^{\frac{2}{5}}}$ .

$\frac{y (x^{2} \cdot 5)}{y^{\frac{2}{5}}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.3

Move $y^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$y (x^{2} \cdot 5) y^{- \frac{2}{5}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.4

Multiply $y$ by $y^{- \frac{2}{5}}$ by adding the exponents.

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Step 6.4.1

Move $y^{- \frac{2}{5}}$ .

$y^{- \frac{2}{5}} y (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.4.2

Multiply $y^{- \frac{2}{5}}$ by $y$ .

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Step 6.4.2.1

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

$y^{- \frac{2}{5}} y^{1} (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.4.2.2

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

$y^{- \frac{2}{5} + 1} (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.4.3

Write $1$ as a fraction with a common denominator.

$y^{- \frac{2}{5} + \frac{5}{5}} (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.4.4

Combine the numerators over the common denominator.

$y^{\frac{- 2 + 5}{5}} (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.4.5

Add $- 2$ and $5$ .

$y^{\frac{3}{5}} (x^{2} \cdot 5) d x + (x^{3} + y^{3}) d y = 0$

Step 6.5

Rewrite using the commutative property of multiplication.

$5 y^{\frac{3}{5}} x^{2} d x + (x^{3} + y^{3}) d y = 0$

Step 6.6

Reorder factors in $5 y^{\frac{3}{5}} x^{2}$ .

$5 x^{2} y^{\frac{3}{5}} d x + (x^{3} + y^{3}) d y = 0$

Step 6.7

Multiply $x^{3} + y^{3}$ by $\frac{1}{y^{\frac{2}{5}}}$ .

$5 x^{2} y^{\frac{3}{5}} d x + (x^{3} + y^{3}) \frac{1}{y^{\frac{2}{5}}} d y = 0$

Step 6.8

Multiply $x^{3} + y^{3}$ by $\frac{1}{y^{\frac{2}{5}}}$ .

$5 x^{2} y^{\frac{3}{5}} d x + \frac{x^{3} + y^{3}}{y^{\frac{2}{5}}} d y = 0$

Step 6.9

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = y$ .

$5 x^{2} y^{\frac{3}{5}} d x + \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}} d y = 0$

Step 7

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

$f (x, y) = \int 5 x^{2} y^{\frac{3}{5}} d x$

Step 8

Integrate $M (x, y) = 5 x^{2} y^{\frac{3}{5}}$ to find $f (x, y)$ .

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Step 8.1

Since $5 y^{\frac{3}{5}}$ is constant with respect to $x$ , move $5 y^{\frac{3}{5}}$ out of the integral.

$f (x, y) = 5 y^{\frac{3}{5}} \int x^{2} d x$

Step 8.2

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

$f (x, y) = 5 y^{\frac{3}{5}} (\frac{1}{3} x^{3} + C)$

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $5 y^{\frac{3}{5}} (\frac{1}{3} x^{3} + C)$ as $5 y^{\frac{3}{5}} \frac{1}{3} x^{3} + C$ .

$f (x, y) = 5 y^{\frac{3}{5}} \frac{1}{3} x^{3} + C$

Step 8.3.2

Simplify.

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Step 8.3.2.1

Combine $5$ and $\frac{1}{3}$ .

$f (x, y) = y^{\frac{3}{5}} \frac{5}{3} x^{3} + C$

Step 8.3.2.2

Combine $y^{\frac{3}{5}}$ and $\frac{5}{3}$ .

$f (x, y) = \frac{y^{\frac{3}{5}} \cdot 5}{3} x^{3} + C$

Step 8.3.2.3

Move $5$ to the left of $y^{\frac{3}{5}}$ .

$f (x, y) = \frac{5 \cdot y^{\frac{3}{5}}}{3} x^{3} + C$

Step 8.3.2.4

Multiply $5$ by $y^{\frac{3}{5}}$ .

$f (x, y) = \frac{5 y^{\frac{3}{5}}}{3} x^{3} + C$

Step 8.3.2.5

Combine $\frac{5 y^{\frac{3}{5}}}{3}$ and $x^{3}$ .

$f (x, y) = \frac{5 y^{\frac{3}{5}} x^{3}}{3} + C$

Step 8.3.3

Reorder terms.

$f (x, y) = \frac{5}{3} y^{\frac{3}{5}} x^{3} + C$

Step 9

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

$f (x, y) = \frac{5}{3} y^{\frac{3}{5}} x^{3} + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

Combine $\frac{5}{3}$ and $y^{\frac{3}{5}}$ .

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

Step 11.3.2

Combine $\frac{5 y^{\frac{3}{5}}}{3}$ and $x^{3}$ .

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

Step 11.3.3

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

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

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{3}{5}$ .

$\frac{5 x^{3}}{3} (\frac{3}{5} y^{\frac{3}{5} - 1}) + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{5 x^{3}}{3} (\frac{3}{5} y^{\frac{3}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.6

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

$\frac{5 x^{3}}{3} (\frac{3}{5} y^{\frac{3}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.7

Combine the numerators over the common denominator.

$\frac{5 x^{3}}{3} (\frac{3}{5} y^{\frac{3 - 1 \cdot 5}{5}}) + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.8

Simplify the numerator.

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Step 11.3.8.1

Multiply $- 1$ by $5$ .

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

Step 11.3.8.2

Subtract $5$ from $3$ .

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

Step 11.3.9

Move the negative in front of the fraction.

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

Step 11.3.10

Combine $\frac{3}{5}$ and $y^{- \frac{2}{5}}$ .

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

Step 11.3.11

Multiply $\frac{5 x^{3}}{3}$ by $\frac{3 y^{- \frac{2}{5}}}{5}$ .

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

Step 11.3.12

Multiply $3$ by $5$ .

$\frac{15 x^{3} y^{- \frac{2}{5}}}{3 \cdot 5} + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.13

Multiply $3$ by $5$ .

$\frac{15 x^{3} y^{- \frac{2}{5}}}{15} + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.14

Move $y^{- \frac{2}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{15 x^{3}}{15 y^{\frac{2}{5}}} + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.15

Cancel the common factor.

$\frac{15 x^{3}}{15 y^{\frac{2}{5}}} + \frac{d}{d y} [g (y)] = \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$

Step 11.3.16

Rewrite the expression.

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Simplify $\frac{d g}{d y} + \frac{x^{3}}{y^{\frac{2}{5}}} - \frac{(x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}}$ .

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Step 12.1.1.1

Combine the numerators over the common denominator.

$\frac{d g}{d y} + \frac{x^{3} - (x + y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Apply the distributive property.

$\frac{d g}{d y} + \frac{x^{3} + (- x - y) (x^{2} - x y + y^{2})}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.2

Expand $(- x - y) (x^{2} - x y + y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{d g}{d y} + \frac{x^{3} - x \cdot x^{2} - x (- x y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3

Simplify each term.

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Step 12.1.1.2.3.1

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

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Step 12.1.1.2.3.1.1

Move $x^{2}$ .

$\frac{d g}{d y} + \frac{x^{3} - (x^{2} x) - x (- x y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.1.2

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

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Step 12.1.1.2.3.1.2.1

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

$\frac{d g}{d y} + \frac{x^{3} - (x^{2} x^{1}) - x (- x y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.1.2.2

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

$\frac{d g}{d y} + \frac{x^{3} - x^{2 + 1} - x (- x y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.1.3

Add $2$ and $1$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - x (- x y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 12.1.1.2.3.2.1

Move $x$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - (x \cdot x) (- 1 y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.2.2

Multiply $x$ by $x$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - x^{2} (- 1 y) - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.3

Rewrite using the commutative property of multiplication.

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - 1 \cdot - 1 x^{2} y - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.4

Multiply $- 1$ by $- 1$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + 1 x^{2} y - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.5

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} - y (- x y) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.6

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

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Step 12.1.1.2.3.6.1

Move $y$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} - (y \cdot y) (- x) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.6.2

Multiply $y$ by $y$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} - y^{2} (- x) - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.7

Rewrite using the commutative property of multiplication.

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} - 1 \cdot - 1 y^{2} x - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.8

Multiply $- 1$ by $- 1$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + 1 y^{2} x - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.9

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + y^{2} x - y \cdot y^{2}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.10

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

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Step 12.1.1.2.3.10.1

Move $y^{2}$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + y^{2} x - (y^{2} y)}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.10.2

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

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Step 12.1.1.2.3.10.2.1

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + y^{2} x - (y^{2} y^{1})}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.10.2.2

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + y^{2} x - y^{2 + 1}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.3.10.3

Add $2$ and $1$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - y x^{2} + y^{2} x - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4

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

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Step 12.1.1.2.4.1

Reorder the factors in the terms $x^{2} y$ and $- y x^{2}$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + x^{2} y - x y^{2} - x^{2} y + y^{2} x - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4.2

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - x y^{2} + 0 + y^{2} x - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4.3

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - x y^{2} + y^{2} x - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4.4

Reorder the factors in the terms $- x y^{2}$ and $y^{2} x$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - y^{2} x + y^{2} x - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4.5

Add $- y^{2} x$ and $y^{2} x$ .

$\frac{d g}{d y} + \frac{x^{3} - x^{3} + 0 - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.2.4.6

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

$\frac{d g}{d y} + \frac{x^{3} - x^{3} - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.3

Simplify by adding terms.

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Step 12.1.1.3.1

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

$\frac{d g}{d y} + \frac{0 - y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.3.2

Subtract $y^{3}$ from $0$ .

$\frac{d g}{d y} + \frac{- y^{3}}{y^{\frac{2}{5}}} = 0$

Step 12.1.1.4

Simplify each term.

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Step 12.1.1.4.1

Move $y^{\frac{2}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{d g}{d y} - y^{3} y^{- \frac{2}{5}} = 0$

Step 12.1.1.4.2

Multiply $y^{3}$ by $y^{- \frac{2}{5}}$ by adding the exponents.

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Step 12.1.1.4.2.1

Move $y^{- \frac{2}{5}}$ .

$\frac{d g}{d y} - (y^{- \frac{2}{5}} y^{3}) = 0$

Step 12.1.1.4.2.2

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

$\frac{d g}{d y} - y^{- \frac{2}{5} + 3} = 0$

Step 12.1.1.4.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{d g}{d y} - y^{- \frac{2}{5} + 3 \cdot \frac{5}{5}} = 0$

Step 12.1.1.4.2.4

Combine $3$ and $\frac{5}{5}$ .

$\frac{d g}{d y} - y^{- \frac{2}{5} + \frac{3 \cdot 5}{5}} = 0$

Step 12.1.1.4.2.5

Combine the numerators over the common denominator.

$\frac{d g}{d y} - y^{\frac{- 2 + 3 \cdot 5}{5}} = 0$

Step 12.1.1.4.2.6

Simplify the numerator.

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Step 12.1.1.4.2.6.1

Multiply $3$ by $5$ .

$\frac{d g}{d y} - y^{\frac{- 2 + 15}{5}} = 0$

Step 12.1.1.4.2.6.2

Add $- 2$ and $15$ .

$\frac{d g}{d y} - y^{\frac{13}{5}} = 0$

Step 12.1.2

Add $y^{\frac{13}{5}}$ to both sides of the equation.

$\frac{d g}{d y} = y^{\frac{13}{5}}$

Step 13

Find the antiderivative of $y^{\frac{13}{5}}$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = y^{\frac{13}{5}}$ .

$\int \frac{d g}{d y} d y = \int y^{\frac{13}{5}} d y$

Step 13.2

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

$g (y) = \int y^{\frac{13}{5}} d y$

Step 13.3

By the Power Rule, the integral of $y^{\frac{13}{5}}$ with respect to $y$ is $\frac{5}{18} y^{\frac{18}{5}}$ .

$g (y) = \frac{5}{18} y^{\frac{18}{5}} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = \frac{5}{3} y^{\frac{3}{5}} x^{3} + g (y)$ .

$f (x, y) = \frac{5}{3} y^{\frac{3}{5}} x^{3} + \frac{5}{18} y^{\frac{18}{5}} + C$

Step 15

Simplify $f (x, y) = \frac{5}{3} y^{\frac{3}{5}} x^{3} + \frac{5}{18} y^{\frac{18}{5}} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

Combine $\frac{5}{3}$ and $y^{\frac{3}{5}}$ .

$f (x, y) = \frac{5 y^{\frac{3}{5}}}{3} x^{3} + \frac{5}{18} y^{\frac{18}{5}} + C$

Step 15.1.2

Combine $\frac{5 y^{\frac{3}{5}}}{3}$ and $x^{3}$ .

$f (x, y) = \frac{5 y^{\frac{3}{5}} x^{3}}{3} + \frac{5}{18} y^{\frac{18}{5}} + C$

Step 15.1.3

Combine $\frac{5}{18}$ and $y^{\frac{18}{5}}$ .

$f (x, y) = \frac{5 y^{\frac{3}{5}} x^{3}}{3} + \frac{5 y^{\frac{18}{5}}}{18} + C$

Step 15.2

Reorder factors in $f (x, y) = \frac{5 y^{\frac{3}{5}} x^{3}}{3} + \frac{5 y^{\frac{18}{5}}}{18} + C$ .

$f (x, y) = \frac{5 x^{3} y^{\frac{3}{5}}}{3} + \frac{5 y^{\frac{18}{5}}}{18} + C$