Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [y^{2}]$

Step 1.4

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

$\frac{\partial M}{\partial y} = 0 + 2 y$

Step 1.5

Add $0$ and $2 y$ .

$\frac{\partial M}{\partial y} = 2 y$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{\partial N}{\partial x} = 3 y \cdot 1$

Step 2.4

Multiply $3$ by $1$ .

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

Step 3

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

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Step 3.1

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

$2 y = 3 y$

Step 3.2

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

$2 y = 3 y$ 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 $2 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{2 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

$\frac{2 y - (3 y)}{N}$

Step 4.3

Substitute $3 x y$ for $N$ .

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Step 4.3.1

Substitute $3 x y$ for $N$ .

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

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $y$ out of $2 y - 1 \cdot 3 y$ .

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Step 4.3.2.1.1

Factor $y$ out of $2 y$ .

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

Step 4.3.2.1.2

Factor $y$ out of $- 1 \cdot 3 y$ .

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

Step 4.3.2.1.3

Factor $y$ out of $y \cdot 2 + y (- 1 \cdot 3)$ .

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

Step 4.3.2.2

Multiply $- 1$ by $3$ .

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

Step 4.3.2.3

Subtract $3$ from $2$ .

$\frac{y \cdot - 1}{3 x y}$

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{y \cdot - 1}{3 x y}$

Step 4.3.3.2

Rewrite the expression.

$\frac{- 1}{3 x}$

Step 4.3.4

Move the negative in front of the fraction.

$- \frac{1}{3 x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{1}{3 x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{1}{3 x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{1}{3 x} d x}$

Step 5.2

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

$μ (x, y) = e^{- (\frac{1}{3} \int \frac{1}{x} d x)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{1}{3} (\ln (| x |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{1}{3} \ln (| x |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{1}{3} \ln (| x |)$ .

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Step 5.5.1.1

Reorder $\ln (| x |)$ and $\frac{1}{3}$ .

$μ (x, y) = e^{- (\frac{1}{3} \ln (| x |))} + C$

Step 5.5.1.2

Simplify $\frac{1}{3} \ln (| x |)$ by moving $\frac{1}{3}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| x |}^{\frac{1}{3}})} + C$

Step 5.5.2

Simplify $- \ln ({| x |}^{\frac{1}{3}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| x |}^{\frac{1}{3}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

$μ (x, y) = {({| x |}^{\frac{1}{3}})}^{- 1} + C$

Step 5.5.4

Multiply the exponents in ${({| x |}^{\frac{1}{3}})}^{- 1}$ .

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Step 5.5.4.1

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

$μ (x, y) = {| x |}^{\frac{1}{3} \cdot - 1} + C$

Step 5.5.4.2

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

$μ (x, y) = {| x |}^{\frac{- 1}{3}} + C$

Step 5.5.4.3

Move the negative in front of the fraction.

$μ (x, y) = {| x |}^{- \frac{1}{3}} + C$

Step 5.5.5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.5

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 6.6

Multiply $x$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 6.6.1

Move $x^{- \frac{1}{3}}$ .

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

Step 6.6.2

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

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Step 6.6.2.1

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

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

Step 6.6.2.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Combine the numerators over the common denominator.

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

Step 6.6.5

Add $- 1$ and $3$ .

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

Step 6.7

Rewrite using the commutative property of multiplication.

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

Step 7

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

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

Step 8

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

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Step 8.1

Since $3 x^{\frac{2}{3}}$ is constant with respect to $y$ , move $3 x^{\frac{2}{3}}$ out of the integral.

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $3 x^{\frac{2}{3}} (\frac{1}{2} y^{2} + C)$ as $3 x^{\frac{2}{3}} \frac{1}{2} y^{2} + C$ .

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 8.3.2.2

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

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

Step 8.3.2.3

Move $3$ to the left of $x^{\frac{2}{3}}$ .

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

Step 8.3.2.4

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

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

Step 8.3.2.5

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

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

Step 8.3.3

Reorder terms.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

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

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

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 11.3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 11.3.6

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

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

Step 11.3.7

Combine the numerators over the common denominator.

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

Step 11.3.8

Simplify the numerator.

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Step 11.3.8.1

Multiply $- 1$ by $3$ .

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

Step 11.3.8.2

Subtract $3$ from $2$ .

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

Step 11.3.9

Move the negative in front of the fraction.

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

Step 11.3.10

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

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

Step 11.3.11

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

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

Step 11.3.12

Multiply $2$ by $3$ .

$\frac{6 y^{2} x^{- \frac{1}{3}}}{2 \cdot 3} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$

Step 11.3.13

Multiply $2$ by $3$ .

$\frac{6 y^{2} x^{- \frac{1}{3}}}{6} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$

Step 11.3.14

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6 y^{2}}{6 x^{\frac{1}{3}}} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$

Step 11.3.15

Cancel the common factor.

$\frac{6 y^{2}}{6 x^{\frac{1}{3}}} + \frac{d}{d x} [g (x)] = \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$

Step 11.3.16

Rewrite the expression.

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

Step 12

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

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Step 12.1

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

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Step 12.1.1

Simplify $\frac{d g}{d x} + \frac{y^{2}}{x^{\frac{1}{3}}} - \frac{x^{2} + y^{2}}{x^{\frac{1}{3}}}$ .

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Step 12.1.1.1

Simplify terms.

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Step 12.1.1.1.1

Combine the numerators over the common denominator.

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

Step 12.1.1.1.2

Apply the distributive property.

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

Step 12.1.1.1.3

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

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Step 12.1.1.1.3.1

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

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

Step 12.1.1.1.3.2

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

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

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Move $x^{\frac{1}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 12.1.1.2.2

Multiply $x^{2}$ by $x^{- \frac{1}{3}}$ by adding the exponents.

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Step 12.1.1.2.2.1

Move $x^{- \frac{1}{3}}$ .

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

Step 12.1.1.2.2.2

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

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

Step 12.1.1.2.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 12.1.1.2.2.4

Combine $2$ and $\frac{3}{3}$ .

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

Step 12.1.1.2.2.5

Combine the numerators over the common denominator.

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

Step 12.1.1.2.2.6

Simplify the numerator.

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Step 12.1.1.2.2.6.1

Multiply $2$ by $3$ .

$\frac{d g}{d x} - x^{\frac{- 1 + 6}{3}} = 0$

Step 12.1.1.2.2.6.2

Add $- 1$ and $6$ .

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

Step 12.1.2

Add $x^{\frac{5}{3}}$ to both sides of the equation.

$\frac{d g}{d x} = x^{\frac{5}{3}}$

Step 13

Find the antiderivative of $x^{\frac{5}{3}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{\frac{5}{3}}$ .

$\int \frac{d g}{d x} d x = \int x^{\frac{5}{3}} d x$

Step 13.2

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

$g (x) = \int x^{\frac{5}{3}} d x$

Step 13.3

By the Power Rule, the integral of $x^{\frac{5}{3}}$ with respect to $x$ is $\frac{3}{8} x^{\frac{8}{3}}$ .

$g (x) = \frac{3}{8} x^{\frac{8}{3}} + C$

Step 14

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

$f (x, y) = \frac{3}{2} x^{\frac{2}{3}} y^{2} + \frac{3}{8} x^{\frac{8}{3}} + C$

Step 15

Simplify $f (x, y) = \frac{3}{2} x^{\frac{2}{3}} y^{2} + \frac{3}{8} x^{\frac{8}{3}} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$f (x, y) = \frac{3 x^{\frac{2}{3}}}{2} y^{2} + \frac{3}{8} x^{\frac{8}{3}} + C$

Step 15.1.2

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

$f (x, y) = \frac{3 x^{\frac{2}{3}} y^{2}}{2} + \frac{3}{8} x^{\frac{8}{3}} + C$

Step 15.1.3

Combine $\frac{3}{8}$ and $x^{\frac{8}{3}}$ .

$f (x, y) = \frac{3 x^{\frac{2}{3}} y^{2}}{2} + \frac{3 x^{\frac{8}{3}}}{8} + C$

Step 15.2

Reorder factors in $f (x, y) = \frac{3 x^{\frac{2}{3}} y^{2}}{2} + \frac{3 x^{\frac{8}{3}}}{8} + C$ .

$f (x, y) = \frac{3 y^{2} x^{\frac{2}{3}}}{2} + \frac{3 x^{\frac{8}{3}}}{8} + C$