Calculus Examples

$(x - 6 y) d x + 5 x d y = 0$

Step 1

Find $\frac{\partial M}{\partial y}$ where $M (x, y) = x - 6 y$ .

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Step 1.1

Differentiate $M$ with respect to $y$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x - 6 y]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x - 6 y$ with respect to $y$ is $\frac{d}{d y} [x] + \frac{d}{d y} [- 6 y]$ .

$\frac{\partial M}{\partial y} = \frac{d}{d y} [x] + \frac{d}{d y} [- 6 y]$

Step 1.2.2

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

$\frac{\partial M}{\partial y} = 0 + \frac{d}{d y} [- 6 y]$

Step 1.3

Evaluate $\frac{d}{d y} [- 6 y]$ .

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Step 1.3.1

Since $- 6$ is constant with respect to $y$ , the derivative of $- 6 y$ with respect to $y$ is $- 6 \frac{d}{d y} [y]$ .

$\frac{\partial M}{\partial y} = 0 - 6 \frac{d}{d y} [y]$

Step 1.3.2

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

$\frac{\partial M}{\partial y} = 0 - 6 \cdot 1$

Step 1.3.3

Multiply $- 6$ by $1$ .

$\frac{\partial M}{\partial y} = 0 - 6$

Step 1.4

Subtract $6$ from $0$ .

$\frac{\partial M}{\partial y} = - 6$

Step 2

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

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Step 2.1

Differentiate $N$ with respect to $x$ .

$\frac{\partial N}{\partial x} = \frac{d}{d x} [5 x]$

Step 2.2

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

$\frac{\partial N}{\partial x} = 5 \frac{d}{d x} [x]$

Step 2.3

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

$\frac{\partial N}{\partial x} = 5 \cdot 1$

Step 2.4

Multiply $5$ by $1$ .

$\frac{\partial N}{\partial x} = 5$

Step 3

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

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Step 3.1

Substitute $- 6$ for $\frac{\partial M}{\partial y}$ and $5$ for $\frac{\partial N}{\partial x}$ .

$- 6 = 5$

Step 3.2

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

$- 6 = 5$ 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 $- 6$ for $\frac{\partial M}{\partial y}$ .

$\frac{- 6 - \frac{\partial N}{\partial x}}{N}$

Step 4.2

Substitute $5$ for $\frac{\partial N}{\partial x}$ .

$\frac{- 6 - 5}{N}$

Step 4.3

Substitute $5 x$ for $N$ .

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Step 4.3.1

Substitute $5 x$ for $N$ .

$\frac{- 6 - 5}{5 x}$

Step 4.3.2

Subtract $5$ from $- 6$ .

$\frac{- 11}{5 x}$

Step 4.3.3

Move the negative in front of the fraction.

$- \frac{11}{5 x}$

Step 4.4

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

$μ (x, y) = e^{\int - \frac{11}{5 x} d x}$

Step 5

Evaluate the integral $e^{\int - \frac{11}{5 x} d x}$ .

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Step 5.1

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

$μ (x, y) = e^{- \int \frac{11}{5 x} d x}$

Step 5.2

Since $\frac{11}{5}$ is constant with respect to $x$ , move $\frac{11}{5}$ out of the integral.

$μ (x, y) = e^{- (\frac{11}{5} \int \frac{1}{x} d x)}$

Step 5.3

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

$μ (x, y) = e^{- \frac{11}{5} (\ln (| x |) + C)}$

Step 5.4

Simplify.

$μ (x, y) = e^{- \frac{11}{5} \ln (| x |)} + C$

Step 5.5

Simplify each term.

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Step 5.5.1

Multiply $- \frac{11}{5} \ln (| x |)$ .

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Step 5.5.1.1

Reorder $\ln (| x |)$ and $\frac{11}{5}$ .

$μ (x, y) = e^{- (\frac{11}{5} \ln (| x |))} + C$

Step 5.5.1.2

Simplify $\frac{11}{5} \ln (| x |)$ by moving $\frac{11}{5}$ inside the logarithm.

$μ (x, y) = e^{- \ln ({| x |}^{\frac{11}{5}})} + C$

Step 5.5.2

Simplify $- \ln ({| x |}^{\frac{11}{5}})$ by moving $- 1$ inside the logarithm.

$μ (x, y) = e^{\ln ({({| x |}^{\frac{11}{5}})}^{- 1})} + C$

Step 5.5.3

Exponentiation and log are inverse functions.

$μ (x, y) = {({| x |}^{\frac{11}{5}})}^{- 1} + C$

Step 5.5.4

Multiply the exponents in ${({| x |}^{\frac{11}{5}})}^{- 1}$ .

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Step 5.5.4.1

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

$μ (x, y) = {| x |}^{\frac{11}{5} \cdot - 1} + C$

Step 5.5.4.2

Multiply $\frac{11}{5} \cdot - 1$ .

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Step 5.5.4.2.1

Combine $\frac{11}{5}$ and $- 1$ .

$μ (x, y) = {| x |}^{\frac{11 \cdot - 1}{5}} + C$

Step 5.5.4.2.2

Multiply $11$ by $- 1$ .

$μ (x, y) = {| x |}^{\frac{- 11}{5}} + C$

Step 5.5.4.3

Move the negative in front of the fraction.

$μ (x, y) = {| x |}^{- \frac{11}{5}} + C$

Step 5.5.5

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

$μ (x, y) = \frac{1}{{| x |}^{\frac{11}{5}}} + C$

Step 6

Multiply both sides of $(x - 6 y) d x + 5 x d y = 0$ by the integration factor $\frac{1}{x^{\frac{11}{5}}}$ .

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Step 6.1

Multiply $x - 6 y$ by $\frac{1}{x^{\frac{11}{5}}}$ .

$(x - 6 y) \frac{1}{x^{\frac{11}{5}}} d x + 5 x d y = 0$

Step 6.2

Multiply $x - 6 y$ by $\frac{1}{x^{\frac{11}{5}}}$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + 5 x d y = 0$

Step 6.3

Multiply $5 x$ by $\frac{1}{x^{\frac{11}{5}}}$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + 5 x \frac{1}{x^{\frac{11}{5}}} d y = 0$

Step 6.4

Multiply $5 x \frac{1}{x^{\frac{11}{5}}}$ .

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Step 6.4.1

Combine $5$ and $\frac{1}{x^{\frac{11}{5}}}$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + x \frac{5}{x^{\frac{11}{5}}} d y = 0$

Step 6.4.2

Combine $x$ and $\frac{5}{x^{\frac{11}{5}}}$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{x \cdot 5}{x^{\frac{11}{5}}} d y = 0$

Step 6.5

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

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11}{5}} x^{- 1}} d y = 0$

Step 6.6

Multiply $x^{\frac{11}{5}}$ by $x^{- 1}$ by adding the exponents.

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Step 6.6.1

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

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11}{5} - 1}} d y = 0$

Step 6.6.2

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

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11}{5} - 1 \cdot \frac{5}{5}}} d y = 0$

Step 6.6.3

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

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11}{5} + \frac{- 1 \cdot 5}{5}}} d y = 0$

Step 6.6.4

Combine the numerators over the common denominator.

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11 - 1 \cdot 5}{5}}} d y = 0$

Step 6.6.5

Simplify the numerator.

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Step 6.6.5.1

Multiply $- 1$ by $5$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{11 - 5}{5}}} d y = 0$

Step 6.6.5.2

Subtract $5$ from $11$ .

$\frac{x - 6 y}{x^{\frac{11}{5}}} d x + \frac{5}{x^{\frac{6}{5}}} d y = 0$

Step 7

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

$f (x, y) = \int \frac{5}{x^{\frac{6}{5}}} d y$

Step 8

Integrate $N (x, y) = \frac{5}{x^{\frac{6}{5}}}$ to find $f (x, y)$ .

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Step 8.1

Apply the constant rule.

$f (x, y) = \frac{5}{x^{\frac{6}{5}}} y + C$

Step 8.2

Combine $\frac{5}{x^{\frac{6}{5}}}$ and $y$ .

$f (x, y) = \frac{5 y}{x^{\frac{6}{5}}} + C$

Step 9

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

$f (x, y) = \frac{5 y}{x^{\frac{6}{5}}} + g (x)$

Step 10

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

$\frac{\partial f}{\partial x} = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11

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

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Step 11.1

Differentiate $f$ with respect to $x$ .

$\frac{d}{d x} [\frac{5 y}{x^{\frac{6}{5}}} + g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.2

By the Sum Rule, the derivative of $\frac{5 y}{x^{\frac{6}{5}}} + g (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{5 y}{x^{\frac{6}{5}}}] + \frac{d}{d x} [g (x)]$ .

$\frac{d}{d x} [\frac{5 y}{x^{\frac{6}{5}}}] + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3

Evaluate $\frac{d}{d x} [\frac{5 y}{x^{\frac{6}{5}}}]$ .

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Step 11.3.1

Since $5 y$ is constant with respect to $x$ , the derivative of $\frac{5 y}{x^{\frac{6}{5}}}$ with respect to $x$ is $5 y \frac{d}{d x} [\frac{1}{x^{\frac{6}{5}}}]$ .

$5 y \frac{d}{d x} [\frac{1}{x^{\frac{6}{5}}}] + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.2

Rewrite $\frac{1}{x^{\frac{6}{5}}}$ as ${(x^{\frac{6}{5}})}^{- 1}$ .

$5 y \frac{d}{d x} [{(x^{\frac{6}{5}})}^{- 1}] + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

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 $g (x) = x^{\frac{6}{5}}$ .

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Step 11.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{6}{5}}$ .

$5 y (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

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 $n = - 1$ .

$5 y (- u^{- 2} \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.3.3

Replace all occurrences of $u$ with $x^{\frac{6}{5}}$ .

$5 y (- {(x^{\frac{6}{5}})}^{- 2} \frac{d}{d x} [x^{\frac{6}{5}}]) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{6}{5}$ .

$5 y (- {(x^{\frac{6}{5}})}^{- 2} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.5

Multiply the exponents in ${(x^{\frac{6}{5}})}^{- 2}$ .

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Step 11.3.5.1

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

$5 y (- x^{\frac{6}{5} \cdot - 2} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.5.2

Multiply $\frac{6}{5} \cdot - 2$ .

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Step 11.3.5.2.1

Combine $\frac{6}{5}$ and $- 2$ .

$5 y (- x^{\frac{6 \cdot - 2}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.5.2.2

Multiply $6$ by $- 2$ .

$5 y (- x^{\frac{- 12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.5.3

Move the negative in front of the fraction.

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.6

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

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} - 1 \cdot \frac{5}{5}})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.7

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

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6}{5} + \frac{- 1 \cdot 5}{5}})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.8

Combine the numerators over the common denominator.

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6 - 1 \cdot 5}{5}})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.9

Simplify the numerator.

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Step 11.3.9.1

Multiply $- 1$ by $5$ .

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{6 - 5}{5}})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.9.2

Subtract $5$ from $6$ .

$5 y (- x^{- \frac{12}{5}} (\frac{6}{5} x^{\frac{1}{5}})) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.10

Combine $\frac{6}{5}$ and $x^{\frac{1}{5}}$ .

$5 y (- x^{- \frac{12}{5}} \frac{6 x^{\frac{1}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.11

Combine $\frac{6 x^{\frac{1}{5}}}{5}$ and $x^{- \frac{12}{5}}$ .

$5 y (- \frac{6 x^{\frac{1}{5}} x^{- \frac{12}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.12

Multiply $x^{\frac{1}{5}}$ by $x^{- \frac{12}{5}}$ by adding the exponents.

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Step 11.3.12.1

Move $x^{- \frac{12}{5}}$ .

$5 y (- \frac{6 (x^{- \frac{12}{5}} x^{\frac{1}{5}})}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.12.2

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

$5 y (- \frac{6 x^{- \frac{12}{5} + \frac{1}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.12.3

Combine the numerators over the common denominator.

$5 y (- \frac{6 x^{\frac{- 12 + 1}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.12.4

Add $- 12$ and $1$ .

$5 y (- \frac{6 x^{\frac{- 11}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.12.5

Move the negative in front of the fraction.

$5 y (- \frac{6 x^{- \frac{11}{5}}}{5}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.13

Move $x^{- \frac{11}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 y (- \frac{6}{5 x^{\frac{11}{5}}}) + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.14

Multiply $- 1$ by $5$ .

$- 5 y \frac{6}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.15

Combine $- 5$ and $\frac{6}{5 x^{\frac{11}{5}}}$ .

$y \frac{- 5 \cdot 6}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.16

Multiply $- 5$ by $6$ .

$y \frac{- 30}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.17

Combine $y$ and $\frac{- 30}{5 x^{\frac{11}{5}}}$ .

$\frac{y \cdot - 30}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.18

Move $- 30$ to the left of $y$ .

$\frac{- 30 \cdot y}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.19

Factor $5$ out of $- 30 y$ .

$\frac{5 (- 6 y)}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.20

Cancel the common factors.

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Step 11.3.20.1

Factor $5$ out of $5 x^{\frac{11}{5}}$ .

$\frac{5 (- 6 y)}{5 (x^{\frac{11}{5}})} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.20.2

Cancel the common factor.

$\frac{5 (- 6 y)}{5 x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.20.3

Rewrite the expression.

$\frac{- 6 y}{x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.3.21

Move the negative in front of the fraction.

$- \frac{6 y}{x^{\frac{11}{5}}} + \frac{d}{d x} [g (x)] = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.4

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

$- \frac{6 y}{x^{\frac{11}{5}}} + \frac{d g}{d x} = \frac{x - 6 y}{x^{\frac{11}{5}}}$

Step 11.5

Reorder terms.

$\frac{d g}{d x} - \frac{6 y}{x^{\frac{11}{5}}} = \frac{x - 6 y}{x^{\frac{11}{5}}}$

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 $\frac{d g}{d x} - \frac{6 y}{x^{\frac{11}{5}}} - \frac{x - 6 y}{x^{\frac{11}{5}}}$ .

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Step 12.1.1.1

Combine the numerators over the common denominator.

$\frac{d g}{d x} + \frac{- 6 y - (x - 6 y)}{x^{\frac{11}{5}}} = 0$

Step 12.1.1.2

Simplify each term.

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Step 12.1.1.2.1

Apply the distributive property.

$\frac{d g}{d x} + \frac{- 6 y - x - (- 6 y)}{x^{\frac{11}{5}}} = 0$

Step 12.1.1.2.2

Multiply $- 6$ by $- 1$ .

$\frac{d g}{d x} + \frac{- 6 y - x + 6 y}{x^{\frac{11}{5}}} = 0$

Step 12.1.1.3

Combine the opposite terms in $- 6 y - x + 6 y$ .

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Step 12.1.1.3.1

Add $- 6 y$ and $6 y$ .

$\frac{d g}{d x} + \frac{- x + 0}{x^{\frac{11}{5}}} = 0$

Step 12.1.1.3.2

Add $- x$ and $0$ .

$\frac{d g}{d x} + \frac{- x}{x^{\frac{11}{5}}} = 0$

Step 12.1.1.4

Simplify each term.

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Step 12.1.1.4.1

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

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11}{5}} x^{- 1}} = 0$

Step 12.1.1.4.2

Multiply $x^{\frac{11}{5}}$ by $x^{- 1}$ by adding the exponents.

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Step 12.1.1.4.2.1

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

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11}{5} - 1}} = 0$

Step 12.1.1.4.2.2

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

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11}{5} - 1 \cdot \frac{5}{5}}} = 0$

Step 12.1.1.4.2.3

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

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11}{5} + \frac{- 1 \cdot 5}{5}}} = 0$

Step 12.1.1.4.2.4

Combine the numerators over the common denominator.

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11 - 1 \cdot 5}{5}}} = 0$

Step 12.1.1.4.2.5

Simplify the numerator.

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Step 12.1.1.4.2.5.1

Multiply $- 1$ by $5$ .

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{11 - 5}{5}}} = 0$

Step 12.1.1.4.2.5.2

Subtract $5$ from $11$ .

$\frac{d g}{d x} + \frac{- 1}{x^{\frac{6}{5}}} = 0$

Step 12.1.1.4.3

Move the negative in front of the fraction.

$\frac{d g}{d x} - \frac{1}{x^{\frac{6}{5}}} = 0$

Step 12.1.2

Add $\frac{1}{x^{\frac{6}{5}}}$ to both sides of the equation.

$\frac{d g}{d x} = \frac{1}{x^{\frac{6}{5}}}$

Step 13

Find the antiderivative of $\frac{1}{x^{\frac{6}{5}}}$ to find $g (x)$ .

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Step 13.1

Integrate both sides of $\frac{d g}{d x} = \frac{1}{x^{\frac{6}{5}}}$ .

$\int \frac{d g}{d x} d x = \int \frac{1}{x^{\frac{6}{5}}} d x$

Step 13.2

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

$g (x) = \int \frac{1}{x^{\frac{6}{5}}} d x$

Step 13.3

Move $x^{\frac{6}{5}}$ out of the denominator by raising it to the $- 1$ power.

$g (x) = \int {(x^{\frac{6}{5}})}^{- 1} d x$

Step 13.4

Multiply the exponents in ${(x^{\frac{6}{5}})}^{- 1}$ .

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Step 13.4.1

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

$g (x) = \int x^{\frac{6}{5} \cdot - 1} d x$

Step 13.4.2

Multiply $\frac{6}{5} \cdot - 1$ .

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Step 13.4.2.1

Combine $\frac{6}{5}$ and $- 1$ .

$g (x) = \int x^{\frac{6 \cdot - 1}{5}} d x$

Step 13.4.2.2

Multiply $6$ by $- 1$ .

$g (x) = \int x^{\frac{- 6}{5}} d x$

Step 13.4.3

Move the negative in front of the fraction.

$g (x) = \int x^{- \frac{6}{5}} d x$

Step 13.5

By the Power Rule, the integral of $x^{- \frac{6}{5}}$ with respect to $x$ is $- 5 x^{- \frac{1}{5}}$ .

$g (x) = - 5 x^{- \frac{1}{5}} + C$

Step 13.6

Simplify the answer.

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Step 13.6.1

Rewrite $- 5 x^{- \frac{1}{5}} + C$ as $- 5 \frac{1}{x^{\frac{1}{5}}} + C$ .

$g (x) = - 5 \frac{1}{x^{\frac{1}{5}}} + C$

Step 13.6.2

Simplify.

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Step 13.6.2.1

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

$g (x) = \frac{- 5}{x^{\frac{1}{5}}} + C$

Step 13.6.2.2

Move the negative in front of the fraction.

$g (x) = - \frac{5}{x^{\frac{1}{5}}} + C$

Step 14

Substitute for $g (x)$ in $f (x, y) = \frac{5 y}{x^{\frac{6}{5}}} + g (x)$ .

$f (x, y) = \frac{5 y}{x^{\frac{6}{5}}} - \frac{5}{x^{\frac{1}{5}}} + C$