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

Differentiate.

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Step 1.2.1

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 1.2.2

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 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 - (2 y)$

Step 1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.4

Subtract $2 y$ from $0$ .

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

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 $- 2$ .

$\frac{y \cdot - 5}{3 x y}$

Step 4.3.3

Cancel the common factor of $y$ .

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Step 4.3.3.1

Cancel the common factor.

$\frac{y \cdot - 5}{3 x y}$

Step 4.3.3.2

Rewrite the expression.

$\frac{- 5}{3 x}$

Step 4.3.4

Move the negative in front of the fraction.

$- \frac{5}{3 x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{5}{3 x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{5}{3 x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{5}{3 x} d x}$

Step 5.2

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

$μ (x, y) = e^{- (\frac{5}{3} \int \frac{1}{x} d x)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{5}{3} (\ln (| x |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{5}{3} \ln (| x |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{5}{3} \ln (| x |)$ .

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Step 5.5.1.1

Reorder $\ln (| x |)$ and $\frac{5}{3}$ .

$μ (x, y) = e^{- (\frac{5}{3} \ln (| x |))} + C$

Step 5.5.1.2

Simplify $\frac{5}{3} \ln (| x |)$ by moving $\frac{5}{3}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| x |}^{\frac{5}{3}})} + C$

Step 5.5.2

Simplify $- \ln ({| x |}^{\frac{5}{3}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| x |}^{\frac{5}{3}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

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

Step 5.5.4

Multiply the exponents in ${({| x |}^{\frac{5}{3}})}^{- 1}$ .

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Step 5.5.4.1

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

$μ (x, y) = {| x |}^{\frac{5}{3} \cdot - 1} + C$

Step 5.5.4.2

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 5.5.4.2.1

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

$μ (x, y) = {| x |}^{\frac{5 \cdot - 1}{3}} + C$

Step 5.5.4.2.2

Multiply $5$ by $- 1$ .

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

Step 5.5.4.3

Move the negative in front of the fraction.

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

Step 5.5.5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

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

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

Step 6.6

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

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

Step 6.7

Multiply $x^{\frac{5}{3}}$ by $x^{- 1}$ by adding the exponents.

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Step 6.7.1

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

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

Step 6.7.2

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

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

Step 6.7.3

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

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

Step 6.7.4

Combine the numerators over the common denominator.

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

Step 6.7.5

Simplify the numerator.

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Step 6.7.5.1

Multiply $- 1$ by $3$ .

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

Step 6.7.5.2

Subtract $3$ from $5$ .

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

Step 6.8

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify the answer.

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Step 8.3.1

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

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

Step 8.3.2

Simplify.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

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

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

Step 8.3.2.4

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

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

Step 9

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

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

Step 10

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

$\frac{\partial f}{\partial x} = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

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

Step 11.2

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

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

Step 11.3

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

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Step 11.3.1

Since $\frac{3 y^{2}}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 y^{2}}{2 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{3 y^{2}}{2} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

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

Step 11.3.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

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

Step 11.3.3

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 $g (x) = x^{\frac{2}{3}}$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

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

Step 11.3.3.2

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

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

Step 11.3.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

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

Step 11.3.4

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

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

Step 11.3.5

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

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Step 11.3.5.1

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

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

Step 11.3.5.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 11.3.5.2.1

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

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

Step 11.3.5.2.2

Multiply $2$ by $- 2$ .

$\frac{3 y^{2}}{2} (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.5.3

Move the negative in front of the fraction.

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.6

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

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.7

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

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.8

Combine the numerators over the common denominator.

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.9

Simplify the numerator.

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Step 11.3.9.1

Multiply $- 1$ by $3$ .

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

Step 11.3.9.2

Subtract $3$ from $2$ .

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.10

Move the negative in front of the fraction.

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}})) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.11

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

$\frac{3 y^{2}}{2} (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3}) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.12

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$\frac{3 y^{2}}{2} (- \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3}) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.13

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 11.3.13.1

Move $x^{- \frac{4}{3}}$ .

$\frac{3 y^{2}}{2} (- \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3}) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.13.2

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

$\frac{3 y^{2}}{2} (- \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3}) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.13.3

Combine the numerators over the common denominator.

$\frac{3 y^{2}}{2} (- \frac{2 x^{\frac{- 4 - 1}{3}}}{3}) + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.13.4

Subtract $1$ from $- 4$ .

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

Step 11.3.13.5

Move the negative in front of the fraction.

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

Step 11.3.14

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.3.15

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

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

Step 11.3.16

Multiply $2$ by $3$ .

$- \frac{6 y^{2}}{2 (3 x^{\frac{5}{3}})} + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.17

Multiply $3$ by $2$ .

$- \frac{6 y^{2}}{6 x^{\frac{5}{3}}} + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.18

Cancel the common factor.

$- \frac{6 y^{2}}{6 x^{\frac{5}{3}}} + \frac{d}{d x} [g (x)] = \frac{(x + y) (x - y)}{x^{\frac{5}{3}}}$

Step 11.3.19

Rewrite the expression.

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

Step 11.4

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

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

Step 11.5

Reorder terms.

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

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

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Step 12.1.1.1

Combine the numerators over the common denominator.

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

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Apply the distributive property.

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

Step 12.1.1.2.2

Expand $(- x - y) (x - y)$ using the FOIL Method.

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Step 12.1.1.2.2.1

Apply the distributive property.

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

Step 12.1.1.2.2.2

Apply the distributive property.

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

Step 12.1.1.2.2.3

Apply the distributive property.

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

Step 12.1.1.2.3

Simplify and combine like terms.

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Step 12.1.1.2.3.1

Simplify each term.

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Step 12.1.1.2.3.1.1

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

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Step 12.1.1.2.3.1.1.1

Move $x$ .

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

Step 12.1.1.2.3.1.1.2

Multiply $x$ by $x$ .

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

Step 12.1.1.2.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 12.1.1.2.3.1.3

Multiply $- 1$ by $- 1$ .

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

Step 12.1.1.2.3.1.4

Multiply $x$ by $1$ .

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

Step 12.1.1.2.3.1.5

Rewrite using the commutative property of multiplication.

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

Step 12.1.1.2.3.1.6

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

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Step 12.1.1.2.3.1.6.1

Move $y$ .

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

Step 12.1.1.2.3.1.6.2

Multiply $y$ by $y$ .

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

Step 12.1.1.2.3.1.7

Multiply $- 1$ by $- 1$ .

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

Step 12.1.1.2.3.1.8

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

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

Step 12.1.1.2.3.2

Subtract $y x$ from $x y$ .

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Step 12.1.1.2.3.2.1

Move $y$ .

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

Step 12.1.1.2.3.2.2

Subtract $x y$ from $x y$ .

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

Step 12.1.1.2.3.3

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

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

Step 12.1.1.3

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

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Step 12.1.1.3.1

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

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

Step 12.1.1.3.2

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

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

Step 12.1.1.4

Simplify each term.

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Step 12.1.1.4.1

Move $x^{\frac{5}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 12.1.1.4.2

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

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Step 12.1.1.4.2.1

Move $x^{- \frac{5}{3}}$ .

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

Step 12.1.1.4.2.2

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

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

Step 12.1.1.4.2.3

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

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

Step 12.1.1.4.2.4

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

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

Step 12.1.1.4.2.5

Combine the numerators over the common denominator.

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

Step 12.1.1.4.2.6

Simplify the numerator.

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Step 12.1.1.4.2.6.1

Multiply $2$ by $3$ .

$\frac{d g}{d x} - x^{\frac{- 5 + 6}{3}} = 0$

Step 12.1.1.4.2.6.2

Add $- 5$ and $6$ .

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

Step 12.1.2

Find a common factor $x^{\frac{1}{3}}$ that is present in each term.

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

Step 12.1.3

Substitute $u$ for $x^{\frac{1}{3}}$ .

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

Step 12.1.4

Solve for $u$ .

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Step 12.1.4.1

Simplify each term.

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Step 12.1.4.1.1

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

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Step 12.1.4.1.1.1

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

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

Step 12.1.4.1.1.2

Cancel the common factor of $3$ .

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Step 12.1.4.1.1.2.1

Cancel the common factor.

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

Step 12.1.4.1.1.2.2

Rewrite the expression.

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

Step 12.1.4.1.2

Simplify.

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

Step 12.1.4.2

Subtract $\frac{d g}{d x}$ from both sides of the equation.

$- u = - \frac{d g}{d x}$

Step 12.1.4.3

Divide each term in $- u = - \frac{d g}{d x}$ by $- 1$ and simplify.

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Step 12.1.4.3.1

Divide each term in $- u = - \frac{d g}{d x}$ by $- 1$ .

$\frac{- u}{- 1} = \frac{- \frac{d g}{d x}}{- 1}$

Step 12.1.4.3.2

Simplify the left side.

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Step 12.1.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{u}{1} = \frac{- \frac{d g}{d x}}{- 1}$

Step 12.1.4.3.2.2

Divide $u$ by $1$ .

$u = \frac{- \frac{d g}{d x}}{- 1}$

Step 12.1.4.3.3

Simplify the right side.

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Step 12.1.4.3.3.1

Dividing two negative values results in a positive value.

$u = \frac{\frac{d g}{d x}}{1}$

Step 12.1.4.3.3.2

Divide $\frac{d g}{d x}$ by $1$ .

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

Step 12.1.5

Substitute $\frac{d g}{d x}$ for $u$ .

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

Step 12.1.6

Rewrite $x^{\frac{1}{3}} = \frac{d g}{d x}$ so $\frac{d g}{d x}$ is on the left side.

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

Step 13

Find the antiderivative of $x^{\frac{1}{3}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = x^{\frac{1}{3}}$ .

$\int \frac{d g}{d x} d x = \int x^{\frac{1}{3}} d x$

Step 13.2

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

$g (x) = \int x^{\frac{1}{3}} d x$

Step 13.3

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

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

Step 14

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

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

Step 15

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

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