Calculus Examples

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

Step 1

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

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Step 1.1

Differentiate $M$ with respect to $y$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Multiply $2$ by $1$ .

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

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $3$ by $3$ .

$\frac{\partial N}{\partial x} = - (9 x^{2} + \frac{d}{d x} [y^{3}])$

Step 2.7

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

$\frac{\partial N}{\partial x} = - (9 x^{2} + 0)$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $9 x^{2}$ and $0$ .

$\frac{\partial N}{\partial x} = - (9 x^{2})$

Step 2.8.2

Multiply $9$ by $- 1$ .

$\frac{\partial N}{\partial x} = - 9 x^{2}$

Step 3

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

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Step 3.1

Substitute $2 x^{2}$ for $\frac{\partial M}{\partial y}$ and $- 9 x^{2}$ for $\frac{\partial N}{\partial x}$ .

$2 x^{2} = - 9 x^{2}$

Step 3.2

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

$2 x^{2} = - 9 x^{2}$ 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 $- 9 x^{2}$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 9 x^{2} - \frac{\partial M}{\partial y}}{M}$

Step 4.2

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

$\frac{- 9 x^{2} - (2 x^{2})}{M}$

Step 4.3

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

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Step 4.3.1

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

$\frac{- 9 x^{2} - (2 x^{2})}{2 x^{2} y}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor $x^{2}$ out of $- 9 x^{2} - 1 \cdot 2 x^{2}$ .

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Step 4.3.2.1.1

Factor $x^{2}$ out of $- 9 x^{2}$ .

$\frac{x^{2} \cdot - 9 - 1 \cdot 2 x^{2}}{2 x^{2} y}$

Step 4.3.2.1.2

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

$\frac{x^{2} \cdot - 9 + x^{2} (- 1 \cdot 2)}{2 x^{2} y}$

Step 4.3.2.1.3

Factor $x^{2}$ out of $x^{2} \cdot - 9 + x^{2} (- 1 \cdot 2)$ .

$\frac{x^{2} (- 9 - 1 \cdot 2)}{2 x^{2} y}$

Step 4.3.2.2

Multiply $- 1$ by $2$ .

$\frac{x^{2} (- 9 - 2)}{2 x^{2} y}$

Step 4.3.2.3

Subtract $2$ from $- 9$ .

$\frac{x^{2} \cdot - 11}{2 x^{2} y}$

Step 4.3.3

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

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Step 4.3.3.1

Cancel the common factor.

$\frac{x^{2} \cdot - 11}{2 x^{2} y}$

Step 4.3.3.2

Rewrite the expression.

$\frac{- 11}{2 y}$

Step 4.3.4

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

$- \frac{11}{2 y}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{11}{2 y} d y}$

Step 5

Evaluate the integral $e^{\int - \frac{11}{2 y} d y}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{11}{2 y} d y}$

Step 5.2

Since $\frac{11}{2}$ is constant with respect to $y$ , move $\frac{11}{2}$ out of the integral.

$μ (x, y) = e^{- (\frac{11}{2} \int \frac{1}{y} d y)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{11}{2} (\ln (| y |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{11}{2} \ln (| y |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{11}{2} \ln (| y |)$ .

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Step 5.5.1.1

Reorder $\ln (| y |)$ and $\frac{11}{2}$ .

$μ (x, y) = e^{- (\frac{11}{2} \ln (| y |))} + C$

Step 5.5.1.2

Simplify $\frac{11}{2} \ln (| y |)$ by moving $\frac{11}{2}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| y |}^{\frac{11}{2}})} + C$

Step 5.5.2

Simplify $- \ln ({| y |}^{\frac{11}{2}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| y |}^{\frac{11}{2}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

$μ (x, y) = {({| y |}^{\frac{11}{2}})}^{- 1} + C$

Step 5.5.4

Multiply the exponents in ${({| y |}^{\frac{11}{2}})}^{- 1}$ .

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Step 5.5.4.1

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

$μ (x, y) = {| y |}^{\frac{11}{2} \cdot - 1} + C$

Step 5.5.4.2

Multiply $\frac{11}{2} \cdot - 1$ .

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Step 5.5.4.2.1

Combine $\frac{11}{2}$ and $- 1$ .

$μ (x, y) = {| y |}^{\frac{11 \cdot - 1}{2}} + C$

Step 5.5.4.2.2

Multiply $11$ by $- 1$ .

$μ (x, y) = {| y |}^{\frac{- 11}{2}} + C$

Step 5.5.4.3

Move the negative in front of the fraction.

$μ (x, y) = {| y |}^{- \frac{11}{2}} + C$

Step 5.5.5

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

$μ (x, y) = \frac{1}{{| y |}^{\frac{11}{2}}} + C$

Step 6

Multiply both sides of $2 x^{2} y d x - (3 x^{3} + y^{3}) d y = 0$ by the integration factor $\frac{1}{y^{\frac{11}{2}}}$ .

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Step 6.1

Multiply $2 x^{2} y$ by $\frac{1}{y^{\frac{11}{2}}}$ .

$2 x^{2} y \frac{1}{y^{\frac{11}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.2

Multiply $2 x^{2} y \frac{1}{y^{\frac{11}{2}}}$ .

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Step 6.2.1

Combine $2$ and $\frac{1}{y^{\frac{11}{2}}}$ .

$x^{2} y \frac{2}{y^{\frac{11}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.2.2

Combine $x^{2}$ and $\frac{2}{y^{\frac{11}{2}}}$ .

$y \frac{x^{2} \cdot 2}{y^{\frac{11}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.2.3

Combine $y$ and $\frac{x^{2} \cdot 2}{y^{\frac{11}{2}}}$ .

$\frac{y (x^{2} \cdot 2)}{y^{\frac{11}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.3

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

$\frac{x^{2} \cdot 2}{y^{\frac{11}{2}} y^{- 1}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4

Multiply $y^{\frac{11}{2}}$ by $y^{- 1}$ by adding the exponents.

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Step 6.4.1

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

$\frac{x^{2} \cdot 2}{y^{\frac{11}{2} - 1}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 2}{y^{\frac{11}{2} - 1 \cdot \frac{2}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4.3

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

$\frac{x^{2} \cdot 2}{y^{\frac{11}{2} + \frac{- 1 \cdot 2}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4.4

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 2}{y^{\frac{11 - 1 \cdot 2}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4.5

Simplify the numerator.

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Step 6.4.5.1

Multiply $- 1$ by $2$ .

$\frac{x^{2} \cdot 2}{y^{\frac{11 - 2}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.4.5.2

Subtract $2$ from $11$ .

$\frac{x^{2} \cdot 2}{y^{\frac{9}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.5

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

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x - (3 x^{3} + y^{3}) d y = 0$

Step 6.6

Multiply $- (3 x^{3} + y^{3})$ by $\frac{1}{y^{\frac{11}{2}}}$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x - (3 x^{3} + y^{3}) \frac{1}{y^{\frac{11}{2}}} d y = 0$

Step 6.7

Apply the distributive property.

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + (- (3 x^{3}) - y^{3}) \frac{1}{y^{\frac{11}{2}}} d y = 0$

Step 6.8

Multiply $3$ by $- 1$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + (- 3 x^{3} - y^{3}) \frac{1}{y^{\frac{11}{2}}} d y = 0$

Step 6.9

Multiply $- 3 x^{3} - y^{3}$ by $\frac{1}{y^{\frac{11}{2}}}$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + \frac{- 3 x^{3} - y^{3}}{y^{\frac{11}{2}}} d y = 0$

Step 6.10

Factor $- 1$ out of $- 3 x^{3}$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + \frac{- (3 x^{3}) - y^{3}}{y^{\frac{11}{2}}} d y = 0$

Step 6.11

Factor $- 1$ out of $- y^{3}$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + \frac{- (3 x^{3}) - (y^{3})}{y^{\frac{11}{2}}} d y = 0$

Step 6.12

Factor $- 1$ out of $- (3 x^{3}) - (y^{3})$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + \frac{- (3 x^{3} + y^{3})}{y^{\frac{11}{2}}} d y = 0$

Step 6.13

Rewrite $- (3 x^{3} + y^{3})$ as $- 1 (3 x^{3} + y^{3})$ .

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x + \frac{- 1 (3 x^{3} + y^{3})}{y^{\frac{11}{2}}} d y = 0$

Step 6.14

Move the negative in front of the fraction.

$\frac{2 x^{2}}{y^{\frac{9}{2}}} d x - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}} d y = 0$

Step 7

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

$f (x, y) = \int \frac{2 x^{2}}{y^{\frac{9}{2}}} d x$

Step 8

Integrate $M (x, y) = \frac{2 x^{2}}{y^{\frac{9}{2}}}$ to find $f (x, y)$ .

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Step 8.1

Since $\frac{2}{y^{\frac{9}{2}}}$ is constant with respect to $x$ , move $\frac{2}{y^{\frac{9}{2}}}$ out of the integral.

$f (x, y) = \frac{2}{y^{\frac{9}{2}}} \int x^{2} d x$

Step 8.2

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

$f (x, y) = \frac{2}{y^{\frac{9}{2}}} (\frac{1}{3} x^{3} + C)$

Step 8.3

Simplify the answer.

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Step 8.3.1

Rewrite $\frac{2}{y^{\frac{9}{2}}} (\frac{1}{3} x^{3} + C)$ as $\frac{2}{y^{\frac{9}{2}}} \cdot \frac{1}{3} x^{3} + C$ .

$f (x, y) = \frac{2}{y^{\frac{9}{2}}} \cdot \frac{1}{3} x^{3} + C$

Step 8.3.2

Simplify.

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Step 8.3.2.1

Multiply $\frac{2}{y^{\frac{9}{2}}}$ by $\frac{1}{3}$ .

$f (x, y) = \frac{2}{y^{\frac{9}{2}} \cdot 3} x^{3} + C$

Step 8.3.2.2

Move $3$ to the left of $y^{\frac{9}{2}}$ .

$f (x, y) = \frac{2}{3 \cdot y^{\frac{9}{2}}} x^{3} + C$

Step 8.3.2.3

Multiply $3$ by $y^{\frac{9}{2}}$ .

$f (x, y) = \frac{2}{3 y^{\frac{9}{2}}} x^{3} + C$

Step 8.3.2.4

Combine $\frac{2}{3 y^{\frac{9}{2}}}$ and $x^{3}$ .

$f (x, y) = \frac{2 x^{3}}{3 y^{\frac{9}{2}}} + C$

Step 9

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

$f (x, y) = \frac{2 x^{3}}{3 y^{\frac{9}{2}}} + g (y)$

Step 10

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

$\frac{\partial f}{\partial y} = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $y$ .

$\frac{d}{d y} [\frac{2 x^{3}}{3 y^{\frac{9}{2}}} + g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.2

By the Sum Rule, the derivative of $\frac{2 x^{3}}{3 y^{\frac{9}{2}}} + g (y)$ with respect to $y$ is $\frac{d}{d y} [\frac{2 x^{3}}{3 y^{\frac{9}{2}}}] + \frac{d}{d y} [g (y)]$ .

$\frac{d}{d y} [\frac{2 x^{3}}{3 y^{\frac{9}{2}}}] + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3

Evaluate $\frac{d}{d y} [\frac{2 x^{3}}{3 y^{\frac{9}{2}}}]$ .

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Step 11.3.1

Since $\frac{2 x^{3}}{3}$ is constant with respect to $y$ , the derivative of $\frac{2 x^{3}}{3 y^{\frac{9}{2}}}$ with respect to $y$ is $\frac{2 x^{3}}{3} \frac{d}{d y} [\frac{1}{y^{\frac{9}{2}}}]$ .

$\frac{2 x^{3}}{3} \frac{d}{d y} [\frac{1}{y^{\frac{9}{2}}}] + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.2

Rewrite $\frac{1}{y^{\frac{9}{2}}}$ as ${(y^{\frac{9}{2}})}^{- 1}$ .

$\frac{2 x^{3}}{3} \frac{d}{d y} [{(y^{\frac{9}{2}})}^{- 1}] + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{- 1}$ and $g (y) = y^{\frac{9}{2}}$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u$ as $y^{\frac{9}{2}}$ .

$\frac{2 x^{3}}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d y} [y^{\frac{9}{2}}]) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

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 $n = - 1$ .

$\frac{2 x^{3}}{3} (- u^{- 2} \frac{d}{d y} [y^{\frac{9}{2}}]) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.3.3

Replace all occurrences of $u$ with $y^{\frac{9}{2}}$ .

$\frac{2 x^{3}}{3} (- {(y^{\frac{9}{2}})}^{- 2} \frac{d}{d y} [y^{\frac{9}{2}}]) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{9}{2}$ .

$\frac{2 x^{3}}{3} (- {(y^{\frac{9}{2}})}^{- 2} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.5

Multiply the exponents in ${(y^{\frac{9}{2}})}^{- 2}$ .

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Step 11.3.5.1

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

$\frac{2 x^{3}}{3} (- y^{\frac{9}{2} \cdot - 2} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.5.2

Cancel the common factor of $2$ .

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Step 11.3.5.2.1

Factor $2$ out of $- 2$ .

$\frac{2 x^{3}}{3} (- y^{\frac{9}{2} \cdot (2 (- 1))} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.5.2.2

Cancel the common factor.

$\frac{2 x^{3}}{3} (- y^{\frac{9}{2} \cdot (2 \cdot - 1)} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.5.2.3

Rewrite the expression.

$\frac{2 x^{3}}{3} (- y^{9 \cdot - 1} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.5.3

Multiply $9$ by $- 1$ .

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{9}{2} - 1})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{9}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.7

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

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.8

Combine the numerators over the common denominator.

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{9 - 1 \cdot 2}{2}})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.9

Simplify the numerator.

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Step 11.3.9.1

Multiply $- 1$ by $2$ .

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{9 - 2}{2}})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.9.2

Subtract $2$ from $9$ .

$\frac{2 x^{3}}{3} (- y^{- 9} (\frac{9}{2} y^{\frac{7}{2}})) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.10

Combine $\frac{9}{2}$ and $y^{\frac{7}{2}}$ .

$\frac{2 x^{3}}{3} (- y^{- 9} \frac{9 y^{\frac{7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.11

Combine $\frac{9 y^{\frac{7}{2}}}{2}$ and $y^{- 9}$ .

$\frac{2 x^{3}}{3} (- \frac{9 y^{\frac{7}{2}} y^{- 9}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12

Multiply $y^{\frac{7}{2}}$ by $y^{- 9}$ by adding the exponents.

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Step 11.3.12.1

Move $y^{- 9}$ .

$\frac{2 x^{3}}{3} (- \frac{9 (y^{- 9} y^{\frac{7}{2}})}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.2

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

$\frac{2 x^{3}}{3} (- \frac{9 y^{- 9 + \frac{7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.3

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 x^{3}}{3} (- \frac{9 y^{- 9 \cdot \frac{2}{2} + \frac{7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.4

Combine $- 9$ and $\frac{2}{2}$ .

$\frac{2 x^{3}}{3} (- \frac{9 y^{\frac{- 9 \cdot 2}{2} + \frac{7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.5

Combine the numerators over the common denominator.

$\frac{2 x^{3}}{3} (- \frac{9 y^{\frac{- 9 \cdot 2 + 7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.6

Simplify the numerator.

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Step 11.3.12.6.1

Multiply $- 9$ by $2$ .

$\frac{2 x^{3}}{3} (- \frac{9 y^{\frac{- 18 + 7}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.6.2

Add $- 18$ and $7$ .

$\frac{2 x^{3}}{3} (- \frac{9 y^{\frac{- 11}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.12.7

Move the negative in front of the fraction.

$\frac{2 x^{3}}{3} (- \frac{9 y^{- \frac{11}{2}}}{2}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.13

Move $y^{- \frac{11}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 x^{3}}{3} (- \frac{9}{2 y^{\frac{11}{2}}}) + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.14

Multiply $\frac{2 x^{3}}{3}$ by $\frac{9}{2 y^{\frac{11}{2}}}$ .

$- \frac{2 x^{3} \cdot 9}{3 (2 y^{\frac{11}{2}})} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.15

Multiply $9$ by $2$ .

$- \frac{18 x^{3}}{3 (2 y^{\frac{11}{2}})} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.16

Multiply $2$ by $3$ .

$- \frac{18 x^{3}}{6 y^{\frac{11}{2}}} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.17

Factor $6$ out of $18 x^{3}$ .

$- \frac{6 (3 x^{3})}{6 y^{\frac{11}{2}}} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.18

Cancel the common factors.

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Step 11.3.18.1

Factor $6$ out of $6 y^{\frac{11}{2}}$ .

$- \frac{6 (3 x^{3})}{6 (y^{\frac{11}{2}})} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.18.2

Cancel the common factor.

$- \frac{6 (3 x^{3})}{6 y^{\frac{11}{2}}} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.3.18.3

Rewrite the expression.

$- \frac{3 x^{3}}{y^{\frac{11}{2}}} + \frac{d}{d y} [g (y)] = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.4

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

$- \frac{3 x^{3}}{y^{\frac{11}{2}}} + \frac{d g}{d y} = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

Step 11.5

Reorder terms.

$\frac{d g}{d y} - \frac{3 x^{3}}{y^{\frac{11}{2}}} = - \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$

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 $\frac{d g}{d y} - \frac{3 x^{3}}{y^{\frac{11}{2}}} + \frac{3 x^{3} + y^{3}}{y^{\frac{11}{2}}}$ .

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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.

$\frac{d g}{d y} + \frac{- 3 x^{3} + 3 x^{3} + y^{3}}{y^{\frac{11}{2}}} = 0$

Step 12.1.1.1.2

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

$\frac{d g}{d y} + \frac{0 + y^{3}}{y^{\frac{11}{2}}} = 0$

Step 12.1.1.1.3

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

$\frac{d g}{d y} + \frac{y^{3}}{y^{\frac{11}{2}}} = 0$

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

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

$\frac{d g}{d y} + \frac{1}{y^{\frac{11}{2}} y^{- 3}} = 0$

Step 12.1.1.2.2

Multiply $y^{\frac{11}{2}}$ by $y^{- 3}$ by adding the exponents.

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Step 12.1.1.2.2.1

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

$\frac{d g}{d y} + \frac{1}{y^{\frac{11}{2} - 3}} = 0$

Step 12.1.1.2.2.2

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

$\frac{d g}{d y} + \frac{1}{y^{\frac{11}{2} - 3 \cdot \frac{2}{2}}} = 0$

Step 12.1.1.2.2.3

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

$\frac{d g}{d y} + \frac{1}{y^{\frac{11}{2} + \frac{- 3 \cdot 2}{2}}} = 0$

Step 12.1.1.2.2.4

Combine the numerators over the common denominator.

$\frac{d g}{d y} + \frac{1}{y^{\frac{11 - 3 \cdot 2}{2}}} = 0$

Step 12.1.1.2.2.5

Simplify the numerator.

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Step 12.1.1.2.2.5.1

Multiply $- 3$ by $2$ .

$\frac{d g}{d y} + \frac{1}{y^{\frac{11 - 6}{2}}} = 0$

Step 12.1.1.2.2.5.2

Subtract $6$ from $11$ .

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

Step 12.1.2

Subtract $\frac{1}{y^{\frac{5}{2}}}$ from both sides of the equation.

$\frac{d g}{d y} = - \frac{1}{y^{\frac{5}{2}}}$

Step 13

Find the antiderivative of $- \frac{1}{y^{\frac{5}{2}}}$ to find $g (y)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d y} = - \frac{1}{y^{\frac{5}{2}}}$ .

$\int \frac{d g}{d y} d y = \int - \frac{1}{y^{\frac{5}{2}}} d y$

Step 13.2

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

$g (y) = \int - \frac{1}{y^{\frac{5}{2}}} d y$

Step 13.3

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

$g (y) = - \int \frac{1}{y^{\frac{5}{2}}} d y$

Step 13.4

Move $y^{\frac{5}{2}}$ out of the denominator by raising it to the $- 1$ power.

$g (y) = - \int {(y^{\frac{5}{2}})}^{- 1} d y$

Step 13.5

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

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Step 13.5.1

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

$g (y) = - \int y^{\frac{5}{2} \cdot - 1} d y$

Step 13.5.2

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

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Step 13.5.2.1

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

$g (y) = - \int y^{\frac{5 \cdot - 1}{2}} d y$

Step 13.5.2.2

Multiply $5$ by $- 1$ .

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

Step 13.5.3

Move the negative in front of the fraction.

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

Step 13.6

By the Power Rule, the integral of $y^{- \frac{5}{2}}$ with respect to $y$ is $- \frac{2}{3} y^{- \frac{3}{2}}$ .

$g (y) = - (- \frac{2}{3} y^{- \frac{3}{2}} + C)$

Step 13.7

Simplify the answer.

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Step 13.7.1

Simplify.

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Step 13.7.1.1

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

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

Step 13.7.1.2

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

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

Step 13.7.1.3

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

$g (y) = - (- \frac{2 y^{- \frac{3}{2}}}{3} + C)$

Step 13.7.1.4

Move $y^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$g (y) = - (- \frac{2}{3 y^{\frac{3}{2}}} + C)$

Step 13.7.2

Simplify.

$g (y) = - - \frac{2}{3 y^{\frac{3}{2}}} + C$

Step 13.7.3

Simplify.

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Step 13.7.3.1

Multiply $- 1$ by $- 1$ .

$g (y) = 1 \frac{2}{3 y^{\frac{3}{2}}} + C$

Step 13.7.3.2

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

$g (y) = \frac{2}{3 y^{\frac{3}{2}}} + C$

Step 14

Substitute for $g (y)$ in $f (x, y) = \frac{2 x^{3}}{3 y^{\frac{9}{2}}} + g (y)$ .

$f (x, y) = \frac{2 x^{3}}{3 y^{\frac{9}{2}}} + \frac{2}{3 y^{\frac{3}{2}}} + C$