Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $2$ .

$\frac{\partial M}{\partial y} = 4 y + \frac{d}{d y} [4 x]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

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

$\frac{\partial M}{\partial y} = 4 y + 0$

Step 1.4.2

Add $4 y$ and $0$ .

$\frac{\partial M}{\partial y} = 4 y$

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 $4 y$ for $\frac{\partial M}{\partial y}$ and $3 y$ for $\frac{\partial N}{\partial x}$ .

$4 y = 3 y$

Step 3.2

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

$4 y = 3 y$ 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 $4 y$ for $\frac{\partial M}{\partial y}$ .

$\frac{4 y - \frac{\partial N}{\partial x}}{N}$

Step 4.2

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

$\frac{4 y - (3 y)}{N}$

Step 4.3

Substitute $3 x y$ for $N$ .

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Step 4.3.1

Substitute $3 x y$ for $N$ .

$\frac{4 y - (3 y)}{3 x y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $y$ out of $4 y - 1 \cdot 3 y$ .

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Step 4.3.2.1.1

Factor $y$ out of $4 y$ .

$\frac{y \cdot 4 - 1 \cdot 3 y}{3 x y}$

Step 4.3.2.1.2

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

$\frac{y \cdot 4 + y (- 1 \cdot 3)}{3 x y}$

Step 4.3.2.1.3

Factor $y$ out of $y \cdot 4 + y (- 1 \cdot 3)$ .

$\frac{y (4 - 1 \cdot 3)}{3 x y}$

Step 4.3.2.2

Multiply $- 1$ by $3$ .

$\frac{y (4 - 3)}{3 x y}$

Step 4.3.2.3

Subtract $3$ from $4$ .

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

Step 4.3.3

Multiply $y$ by $1$ .

$\frac{y}{3 x y}$

Step 4.3.4

Cancel the common factor of $y$ .

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Step 4.3.4.1

Cancel the common factor.

$\frac{y}{3 x y}$

Step 4.3.4.2

Rewrite the expression.

$\frac{1}{3 x}$

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 $\frac{1}{3}$ 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 5.2

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

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

Step 5.3

Simplify.

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

Step 5.4

Simplify each term.

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Step 5.4.1

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 5.4.2

Exponentiation and log are inverse functions.

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

Step 6

Multiply both sides of $(2 y^{2} + 4 x) d x + 3 x y d y = 0$ by the integration factor $x^{\frac{1}{3}}$ .

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Step 6.1

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

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

Step 6.2

Apply the distributive property.

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

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

Step 6.3.3

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

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

Step 6.3.4

Combine the numerators over the common denominator.

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

Step 6.3.5

Add $1$ and $3$ .

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

Step 6.4

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

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

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

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Step 6.5.2.1

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

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

Step 6.5.2.2

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

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

Step 6.5.3

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

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

Step 6.5.4

Combine the numerators over the common denominator.

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

Step 6.5.5

Add $1$ and $3$ .

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

Step 6.6

Reorder factors in $3 x^{\frac{4}{3}} y$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

Since $3 x^{\frac{4}{3}}$ is constant with respect to $y$ , move $3 x^{\frac{4}{3}}$ out of the integral.

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $3 x^{\frac{4}{3}} (\frac{1}{2} y^{2} + C)$ as $3 x^{\frac{4}{3}} \frac{1}{2} y^{2} + C$ .

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

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

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

Step 8.3.2.4

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

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

Step 8.3.2.5

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

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

Step 8.3.3

Reorder terms.

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

Evaluate $\frac{d}{d x} [\frac{3}{2} x^{\frac{4}{3}} y^{2}]$ .

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Step 11.3.1

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

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

Step 11.3.2

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

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

Step 11.3.3

Since $\frac{3 y^{2}}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 x^{\frac{4}{3}} y^{2}}{2}$ with respect to $x$ is $\frac{3 y^{2}}{2} \frac{d}{d x} [x^{\frac{4}{3}}]$ .

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

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

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

Step 11.3.5

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

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

Step 11.3.6

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

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

Step 11.3.7

Combine the numerators over the common denominator.

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

Step 11.3.8

Simplify the numerator.

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Step 11.3.8.1

Multiply $- 1$ by $3$ .

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

Step 11.3.8.2

Subtract $3$ from $4$ .

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

Step 11.3.9

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

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

Step 11.3.10

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

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

Step 11.3.11

Multiply $4$ by $3$ .

$\frac{12 y^{2} x^{\frac{1}{3}}}{2 \cdot 3} + \frac{d}{d x} [g (x)] = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11.3.12

Multiply $2$ by $3$ .

$\frac{12 y^{2} x^{\frac{1}{3}}}{6} + \frac{d}{d x} [g (x)] = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11.3.13

Factor $6$ out of $12 y^{2} x^{\frac{1}{3}}$ .

$\frac{6 (2 y^{2} x^{\frac{1}{3}})}{6} + \frac{d}{d x} [g (x)] = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11.3.14

Cancel the common factors.

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Step 11.3.14.1

Factor $6$ out of $6$ .

$\frac{6 (2 y^{2} x^{\frac{1}{3}})}{6 (1)} + \frac{d}{d x} [g (x)] = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11.3.14.2

Cancel the common factor.

$\frac{6 (2 y^{2} x^{\frac{1}{3}})}{6 \cdot 1} + \frac{d}{d x} [g (x)] = 2 y^{2} x^{\frac{1}{3}} + 4 x^{\frac{4}{3}}$

Step 11.3.14.3

Rewrite the expression.

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

Step 11.3.14.4

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

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

Step 11.4

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

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

Step 11.5

Simplify.

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Step 11.5.1

Reorder terms.

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

Step 11.5.2

Reorder factors in $\frac{d g}{d x} + 2 x^{\frac{1}{3}} y^{2}$ .

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

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

Combine the opposite terms in $\frac{d g}{d x} + 2 y^{2} x^{\frac{1}{3}} - 2 y^{2} x^{\frac{1}{3}} - 4 x^{\frac{4}{3}}$ .

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Step 12.1.1.1

Subtract $2 y^{2} x^{\frac{1}{3}}$ from $2 y^{2} x^{\frac{1}{3}}$ .

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

Step 12.1.1.2

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

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

Step 12.1.2

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

$\frac{d g}{d x} = 4 x^{\frac{4}{3}}$

Step 13

Find the antiderivative of $4 x^{\frac{4}{3}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = 4 x^{\frac{4}{3}}$ .

$\int \frac{d g}{d x} d x = \int 4 x^{\frac{4}{3}} d x$

Step 13.2

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

$g (x) = \int 4 x^{\frac{4}{3}} d x$

Step 13.3

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

$g (x) = 4 \int x^{\frac{4}{3}} d x$

Step 13.4

By the Power Rule, the integral of $x^{\frac{4}{3}}$ with respect to $x$ is $\frac{3}{7} x^{\frac{7}{3}}$ .

$g (x) = 4 (\frac{3}{7} x^{\frac{7}{3}} + C)$

Step 13.5

Simplify the answer.

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Step 13.5.1

Rewrite $4 (\frac{3}{7} x^{\frac{7}{3}} + C)$ as $4 (\frac{3}{7}) x^{\frac{7}{3}} + C$ .

$g (x) = 4 (\frac{3}{7}) x^{\frac{7}{3}} + C$

Step 13.5.2

Simplify.

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Step 13.5.2.1

Combine $4$ and $\frac{3}{7}$ .

$g (x) = \frac{4 \cdot 3}{7} x^{\frac{7}{3}} + C$

Step 13.5.2.2

Multiply $4$ by $3$ .

$g (x) = \frac{12}{7} x^{\frac{7}{3}} + C$

Step 14

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

$f (x, y) = \frac{3}{2} x^{\frac{4}{3}} y^{2} + \frac{12}{7} x^{\frac{7}{3}} + C$

Step 15

Simplify $f (x, y) = \frac{3}{2} x^{\frac{4}{3}} y^{2} + \frac{12}{7} x^{\frac{7}{3}} + C$ .

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Step 15.1

Simplify each term.

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Step 15.1.1

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

$f (x, y) = \frac{3 x^{\frac{4}{3}}}{2} y^{2} + \frac{12}{7} x^{\frac{7}{3}} + C$

Step 15.1.2

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

$f (x, y) = \frac{3 x^{\frac{4}{3}} y^{2}}{2} + \frac{12}{7} x^{\frac{7}{3}} + C$

Step 15.1.3

Combine $\frac{12}{7}$ and $x^{\frac{7}{3}}$ .

$f (x, y) = \frac{3 x^{\frac{4}{3}} y^{2}}{2} + \frac{12 x^{\frac{7}{3}}}{7} + C$

Step 15.2

Reorder factors in $f (x, y) = \frac{3 x^{\frac{4}{3}} y^{2}}{2} + \frac{12 x^{\frac{7}{3}}}{7} + C$ .

$f (x, y) = \frac{3 y^{2} x^{\frac{4}{3}}}{2} + \frac{12 x^{\frac{7}{3}}}{7} + C$