Calculus Examples

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

Step 1

Let $v = e^{6 x - 5 y}$ . Substitute $v$ for all occurrences of $e^{6 x - 5 y}$ .

$\frac{d y}{d x} = x v$

Step 2

Find $\frac{d v}{d x}$ by differentiating $e^{6 x - 5 y}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 6 x - 5 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $6 x - 5 y$ .

$\frac{d v}{d x} = \frac{d}{d u} [e^{u}] \frac{d}{d x} [6 x - 5 y]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$\frac{d v}{d x} = e^{u} \frac{d}{d x} [6 x - 5 y]$

Step 2.1.3

Replace all occurrences of $u$ with $6 x - 5 y$ .

$\frac{d v}{d x} = e^{6 x - 5 y} \frac{d}{d x} [6 x - 5 y]$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $6$ by $1$ .

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

Step 2.6

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

$\frac{d v}{d x} = e^{6 x - 5 y} (6 - 5 \frac{d}{d x} [y])$

Step 2.7

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{d v}{d x} = e^{6 x - 5 y} (6 - 5 y')$

Step 3

Substitute $v$ for $e^{6 x - 5 y}$ .

$\frac{d v}{d x} = v (6 - 5 y')$

Step 4

Substitute the derivative back in to the differential equation.

$- \frac{\frac{d v}{d x}}{5 v} + \frac{6}{5} = x v$

Step 5

Multiply each term in $- \frac{\frac{d v}{d x}}{5 v} + \frac{6}{5} = x v$ by $- (5 v)$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{\frac{d v}{d x}}{5 v} + \frac{6}{5} = x v$ by $- (5 v)$ .

$- \frac{\frac{d v}{d x}}{5 v} (- (5 v)) + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of $5 v$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{5 v}$ into the numerator.

$\frac{- \frac{d v}{d x}}{5 v} (- (5 v)) + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.1.2

Factor $5 v$ out of $- (5 v)$ .

$\frac{- \frac{d v}{d x}}{5 v} (5 v (- 1)) + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{5 v} (5 v \cdot - 1) + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

$\frac{d v}{d x} + \frac{6}{5} (- (5 v)) = x v (- (5 v))$

Step 5.2.1.4

Cancel the common factor of $5$ .

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Step 5.2.1.4.1

Factor $5$ out of $- (5 v)$ .

$\frac{d v}{d x} + \frac{6}{5} (5 (- (v))) = x v (- (5 v))$

Step 5.2.1.4.2

Cancel the common factor.

$\frac{d v}{d x} + \frac{6}{5} (5 (- (v))) = x v (- (5 v))$

Step 5.2.1.4.3

Rewrite the expression.

$\frac{d v}{d x} + 6 (- (v)) = x v (- (5 v))$

Step 5.2.1.5

Multiply $- 1$ by $6$ .

$\frac{d v}{d x} - 6 v = x v (- (5 v))$

Step 5.3

Simplify the right side.

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Step 5.3.1

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - 6 v = - 1 \cdot 5 x v \cdot v$

Step 5.3.2

Multiply $v$ by $v$ by adding the exponents.

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Step 5.3.2.1

Move $v$ .

$\frac{d v}{d x} - 6 v = - 1 \cdot 5 x (v \cdot v)$

Step 5.3.2.2

Multiply $v$ by $v$ .

$\frac{d v}{d x} - 6 v = - 1 \cdot 5 x v^{2}$

Step 5.3.3

Multiply $- 1$ by $5$ .

$\frac{d v}{d x} - 6 v = - 5 x v^{2}$

Step 6

To solve the differential equation, let $v = y^{1 - n}$ where $n$ is the exponent of $v^{2}$ .

$v = v^{- 1}$

Step 7

Solve the equation for $v$ .

$y = v^{- 1}$

Step 8

Take the derivative of $y$ with respect to $x$ .

$y' = v^{- 1}$

Step 9

Take the derivative of $v^{- 1}$ with respect to $x$ .

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Step 9.1

Take the derivative of $v^{- 1}$ .

$y' = \frac{d}{d x} [v^{- 1}]$

Step 9.2

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

$y' = \frac{d}{d x} [\frac{1}{v}]$

Step 9.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1$ and $g (x) = v$ .

$y' = \frac{v \frac{d}{d x} [1] - 1 \cdot 1 \frac{d}{d x} [v]}{v^{2}}$

Step 9.4

Differentiate using the Constant Rule.

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Step 9.4.1

Multiply $- 1$ by $1$ .

$y' = \frac{v \frac{d}{d x} [1] - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$y' = \frac{v \cdot 0 - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3

Simplify the expression.

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Step 9.4.3.1

Multiply $v$ by $0$ .

$y' = \frac{0 - \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3.2

Subtract $\frac{d}{d x} [v]$ from $0$ .

$y' = \frac{- \frac{d}{d x} [v]}{v^{2}}$

Step 9.4.3.3

Move the negative in front of the fraction.

$y' = - \frac{\frac{d}{d x} [v]}{v^{2}}$

Step 9.5

Rewrite $\frac{d}{d x} [v]$ as $v'$ .

$y' = - \frac{v'}{v^{2}}$

Step 10

Substitute $- \frac{v'}{v^{2}}$ for $\frac{d v}{d x}$ and $v^{- 1}$ for $v$ in the original equation $\frac{d v}{d x} - 6 v = - 5 x v^{2}$ .

$- \frac{v'}{v^{2}} - 6 v^{- 1} = - 5 x {(v^{- 1})}^{2}$

Step 11

Solve the substituted differential equation.

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Step 11.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - 6 v^{- 1} = - 5 x {(v^{- 1})}^{2}$ by $- v^{2}$ to eliminate the fractions.

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Step 11.1.1

Multiply each term in $- \frac{\frac{d v}{d x}}{v^{2}} - 6 v^{- 1} = - 5 x {(v^{- 1})}^{2}$ by $- v^{2}$ .

$- \frac{\frac{d v}{d x}}{v^{2}} (- v^{2}) - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2

Simplify the left side.

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Step 11.1.2.1

Simplify each term.

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Step 11.1.2.1.1

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

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Step 11.1.2.1.1.1

Move the leading negative in $- \frac{\frac{d v}{d x}}{v^{2}}$ into the numerator.

$\frac{- \frac{d v}{d x}}{v^{2}} (- v^{2}) - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.1.2

Factor $v^{2}$ out of $- v^{2}$ .

$\frac{- \frac{d v}{d x}}{v^{2}} (v^{2} \cdot - 1) - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.1.3

Cancel the common factor.

$\frac{- \frac{d v}{d x}}{v^{2}} (v^{2} \cdot - 1) - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.1.4

Rewrite the expression.

$- \frac{d v}{d x} \cdot - 1 - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.2

Multiply $- 1$ by $- 1$ .

$1 \frac{d v}{d x} - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.3

Multiply $\frac{d v}{d x}$ by $1$ .

$\frac{d v}{d x} - 6 v^{- 1} (- v^{2}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{d v}{d x} - 6 \cdot - 1 v^{- 1} v^{2} = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.5

Multiply $v^{- 1}$ by $v^{2}$ by adding the exponents.

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Step 11.1.2.1.5.1

Move $v^{2}$ .

$\frac{d v}{d x} - 6 \cdot - 1 (v^{2} v^{- 1}) = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.5.2

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

$\frac{d v}{d x} - 6 \cdot - 1 v^{2 - 1} = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.5.3

Subtract $1$ from $2$ .

$\frac{d v}{d x} - 6 \cdot - 1 v^{1} = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.6

Simplify $- 6 \cdot - 1 v^{1}$ .

$\frac{d v}{d x} - 6 \cdot - 1 v = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.2.1.7

Multiply $- 6$ by $- 1$ .

$\frac{d v}{d x} + 6 v = - 5 x {(v^{- 1})}^{2} (- v^{2})$

Step 11.1.3

Simplify the right side.

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Step 11.1.3.1

Multiply the exponents in ${(v^{- 1})}^{2}$ .

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Step 11.1.3.1.1

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

$\frac{d v}{d x} + 6 v = - 5 x v^{- 1 \cdot 2} (- v^{2})$

Step 11.1.3.1.2

Multiply $- 1$ by $2$ .

$\frac{d v}{d x} + 6 v = - 5 x v^{- 2} (- v^{2})$

Step 11.1.3.2

Multiply $v^{- 2}$ by $v^{2}$ by adding the exponents.

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Step 11.1.3.2.1

Move $v^{2}$ .

$\frac{d v}{d x} + 6 v = - 5 x (v^{2} v^{- 2}) \cdot - 1$

Step 11.1.3.2.2

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

$\frac{d v}{d x} + 6 v = - 5 x v^{2 - 2} \cdot - 1$

Step 11.1.3.2.3

Subtract $2$ from $2$ .

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

Step 11.1.3.3

Simplify $- 5 x v^{0} \cdot - 1$ .

$\frac{d v}{d x} + 6 v = - 5 x \cdot - 1$

Step 11.1.3.4

Multiply $- 1$ by $- 5$ .

$\frac{d v}{d x} + 6 v = 5 x$

Step 11.2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 6$ .

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Step 11.2.1

Set up the integration.

$e^{\int 6 d x}$

Step 11.2.2

Apply the constant rule.

$e^{6 x + C}$

Step 11.2.3

Remove the constant of integration.

$e^{6 x}$

Step 11.3

Multiply each term by the integrating factor $e^{6 x}$ .

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Step 11.3.1

Multiply each term by $e^{6 x}$ .

$e^{6 x} \frac{d v}{d x} + e^{6 x} (6 v) = e^{6 x} (5 x)$

Step 11.3.2

Rewrite using the commutative property of multiplication.

$e^{6 x} \frac{d v}{d x} + 6 e^{6 x} v = e^{6 x} (5 x)$

Step 11.3.3

Rewrite using the commutative property of multiplication.

$e^{6 x} \frac{d v}{d x} + 6 e^{6 x} v = 5 e^{6 x} x$

Step 11.3.4

Reorder factors in $e^{6 x} \frac{d v}{d x} + 6 e^{6 x} v = 5 e^{6 x} x$ .

$e^{6 x} \frac{d v}{d x} + 6 v e^{6 x} = 5 x e^{6 x}$

Step 11.4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{6 x} v] = 5 x e^{6 x}$

Step 11.5

Set up an integral on each side.

$\int \frac{d}{d x} [e^{6 x} v] d x = \int 5 x e^{6 x} d x$

Step 11.6

Integrate the left side.

$e^{6 x} v = \int 5 x e^{6 x} d x$

Step 11.7

Integrate the right side.

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Step 11.7.1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$e^{6 x} v = 5 \int x e^{6 x} d x$

Step 11.7.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{6 x}$ .

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

Step 11.7.3

Simplify.

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Step 11.7.3.1

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

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

Step 11.7.3.2

Combine $x$ and $\frac{e^{6 x}}{6}$ .

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

Step 11.7.3.3

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

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

Step 11.7.4

Since $\frac{1}{6}$ is constant with respect to $x$ , move $\frac{1}{6}$ out of the integral.

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

Step 11.7.5

Let $u = 6 x$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.7.5.1

Let $u = 6 x$ . Find $\frac{d u}{d x}$ .

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Step 11.7.5.1.1

Differentiate $6 x$ .

$\frac{d}{d x} [6 x]$

Step 11.7.5.1.2

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

$6 \frac{d}{d x} [x]$

Step 11.7.5.1.3

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

$6 \cdot 1$

Step 11.7.5.1.4

Multiply $6$ by $1$ .

$6$

Step 11.7.5.2

Rewrite the problem using $u$ and $d u$ .

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

Step 11.7.6

Combine $e^{u}$ and $\frac{1}{6}$ .

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

Step 11.7.7

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

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

Step 11.7.8

Simplify.

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Step 11.7.8.1

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

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

Step 11.7.8.2

Multiply $6$ by $6$ .

$e^{6 x} v = 5 (\frac{x e^{6 x}}{6} - \frac{1}{36} \int e^{u} d u)$

Step 11.7.9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{6 x} v = 5 (\frac{x e^{6 x}}{6} - \frac{1}{36} (e^{u} + C))$

Step 11.7.10

Rewrite $5 (\frac{x e^{6 x}}{6} - \frac{1}{36} (e^{u} + C))$ as $5 (\frac{1}{6} x e^{6 x} - \frac{1}{36} e^{u}) + C$ .

$e^{6 x} v = 5 (\frac{1}{6} x e^{6 x} - \frac{1}{36} e^{u}) + C$

Step 11.7.11

Replace all occurrences of $u$ with $6 x$ .

$e^{6 x} v = 5 (\frac{1}{6} x e^{6 x} - \frac{1}{36} e^{6 x}) + C$

Step 11.8

Solve for $v$ .

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Step 11.8.1

Simplify.

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Step 11.8.1.1

Combine $e^{6 x}$ and $\frac{1}{36}$ .

$e^{6 x} v = 5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36}) + C$

Step 11.8.1.2

Remove parentheses.

$e^{6 x} v = 5 (\frac{1}{6} \cdot x e^{6 x} - \frac{e^{6 x}}{36}) + C$

Step 11.8.2

Divide each term in $e^{6 x} v = 5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36}) + C$ by $e^{6 x}$ and simplify.

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Step 11.8.2.1

Divide each term in $e^{6 x} v = 5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36}) + C$ by $e^{6 x}$ .

$\frac{e^{6 x} v}{e^{6 x}} = \frac{5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.2

Simplify the left side.

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Step 11.8.2.2.1

Cancel the common factor of $e^{6 x}$ .

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Step 11.8.2.2.1.1

Cancel the common factor.

$\frac{e^{6 x} v}{e^{6 x}} = \frac{5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.2.1.2

Divide $v$ by $1$ .

$v = \frac{5 (\frac{1}{6} \cdot (x e^{6 x}) - \frac{e^{6 x}}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3

Simplify the right side.

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Step 11.8.2.3.1

Simplify each term.

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Step 11.8.2.3.1.1

Simplify the numerator.

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Step 11.8.2.3.1.1.1

Factor $e^{6 x}$ out of $\frac{1}{6} \cdot x e^{6 x} - \frac{e^{6 x}}{36}$ .

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Step 11.8.2.3.1.1.1.1

Factor $e^{6 x}$ out of $\frac{1}{6} \cdot x e^{6 x}$ .

$v = \frac{5 (e^{6 x} (\frac{1}{6} \cdot x) - \frac{e^{6 x}}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.1.2

Factor $e^{6 x}$ out of $- \frac{e^{6 x}}{36}$ .

$v = \frac{5 (e^{6 x} (\frac{1}{6} \cdot x) + e^{6 x} (- \frac{1}{36}))}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.1.3

Factor $e^{6 x}$ out of $e^{6 x} (\frac{1}{6} \cdot x) + e^{6 x} (- \frac{1}{36})$ .

$v = \frac{5 (e^{6 x} (\frac{1}{6} \cdot x - \frac{1}{36}))}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.2

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

$v = \frac{5 e^{6 x} (\frac{x}{6} - \frac{1}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.3

To write $\frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$v = \frac{5 e^{6 x} (\frac{x}{6} \cdot \frac{6}{6} - \frac{1}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.4

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.8.2.3.1.1.4.1

Multiply $\frac{x}{6}$ by $\frac{6}{6}$ .

$v = \frac{5 e^{6 x} (\frac{x \cdot 6}{6 \cdot 6} - \frac{1}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.4.2

Multiply $6$ by $6$ .

$v = \frac{5 e^{6 x} (\frac{x \cdot 6}{36} - \frac{1}{36})}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.5

Combine the numerators over the common denominator.

$v = \frac{5 e^{6 x} \frac{x \cdot 6 - 1}{36}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.6

Move $6$ to the left of $x$ .

$v = \frac{5 e^{6 x} \frac{6 x - 1}{36}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.7

Combine exponents.

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Step 11.8.2.3.1.1.7.1

Combine $\frac{6 x - 1}{36}$ and $5$ .

$v = \frac{\frac{(6 x - 1) \cdot 5}{36} e^{6 x}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.7.2

Combine $\frac{(6 x - 1) \cdot 5}{36}$ and $e^{6 x}$ .

$v = \frac{\frac{(6 x - 1) \cdot 5 e^{6 x}}{36}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.1.8

Move $5$ to the left of $6 x - 1$ .

$v = \frac{\frac{5 (6 x - 1) e^{6 x}}{36}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$v = \frac{5 (6 x - 1) e^{6 x}}{36} \cdot \frac{1}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.3

Combine.

$v = \frac{5 (6 x - 1) e^{6 x} \cdot 1}{36 e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.4

Cancel the common factor of $e^{6 x}$ .

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Step 11.8.2.3.1.4.1

Cancel the common factor.

$v = \frac{5 (6 x - 1) e^{6 x} \cdot 1}{36 e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.4.2

Rewrite the expression.

$v = \frac{5 (6 x - 1) \cdot 1}{36} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.1.5

Multiply $5$ by $1$ .

$v = \frac{5 (6 x - 1)}{36} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.2

To write $\frac{5 (6 x - 1)}{36}$ as a fraction with a common denominator, multiply by $\frac{e^{6 x}}{e^{6 x}}$ .

$v = \frac{5 (6 x - 1)}{36} \cdot \frac{e^{6 x}}{e^{6 x}} + \frac{C}{e^{6 x}}$

Step 11.8.2.3.3

To write $\frac{C}{e^{6 x}}$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

$v = \frac{5 (6 x - 1)}{36} \cdot \frac{e^{6 x}}{e^{6 x}} + \frac{C}{e^{6 x}} \cdot \frac{36}{36}$

Step 11.8.2.3.4

Write each expression with a common denominator of $36 e^{6 x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.8.2.3.4.1

Multiply $\frac{5 (6 x - 1)}{36}$ by $\frac{e^{6 x}}{e^{6 x}}$ .

$v = \frac{5 (6 x - 1) e^{6 x}}{36 e^{6 x}} + \frac{C}{e^{6 x}} \cdot \frac{36}{36}$

Step 11.8.2.3.4.2

Multiply $\frac{C}{e^{6 x}}$ by $\frac{36}{36}$ .

$v = \frac{5 (6 x - 1) e^{6 x}}{36 e^{6 x}} + \frac{C \cdot 36}{e^{6 x} \cdot 36}$

Step 11.8.2.3.4.3

Reorder the factors of $e^{6 x} \cdot 36$ .

$v = \frac{5 (6 x - 1) e^{6 x}}{36 e^{6 x}} + \frac{C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.5

Combine the numerators over the common denominator.

$v = \frac{5 (6 x - 1) e^{6 x} + C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.6

Simplify the numerator.

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Step 11.8.2.3.6.1

Apply the distributive property.

$v = \frac{(5 (6 x) + 5 \cdot - 1) e^{6 x} + C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.6.2

Multiply $6$ by $5$ .

$v = \frac{(30 x + 5 \cdot - 1) e^{6 x} + C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.6.3

Multiply $5$ by $- 1$ .

$v = \frac{(30 x - 5) e^{6 x} + C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.6.4

Apply the distributive property.

$v = \frac{30 x e^{6 x} - 5 e^{6 x} + C \cdot 36}{36 e^{6 x}}$

Step 11.8.2.3.6.5

Move $36$ to the left of $C$ .

$v = \frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}}$

Step 12

Substitute $v^{- 1}$ for $v$ .

$v^{- 1} = \frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}}$

Step 13

Replace all occurrences of $v$ with $e^{6 x - 5 y}$ .

${(e^{6 x - 5 y})}^{- 1} = \frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}}$

Step 14

Solve for $y$ .

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Step 14.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln ({(e^{6 x - 5 y})}^{- 1}) = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$

Step 14.2

Expand the left side.

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Step 14.2.1

Expand $\ln ({(e^{6 x - 5 y})}^{- 1})$ by moving $- 1$ outside the logarithm.

$- \ln (e^{6 x - 5 y}) = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$

Step 14.2.2

Expand $\ln (e^{6 x - 5 y})$ by moving $6 x - 5 y$ outside the logarithm.

$- ((6 x - 5 y) \ln (e)) = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$

Step 14.2.3

The natural logarithm of $e$ is $1$ .

$- ((6 x - 5 y) \cdot 1) = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$

Step 14.2.4

Multiply $6 x - 5 y$ by $1$ .

$- (6 x - 5 y) = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$

Step 14.3

Expand the right side.

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Step 14.3.1

Rewrite $\ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36 e^{6 x}})$ as $\ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - \ln (36 e^{6 x})$ .

$- (6 x - 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - \ln (36 e^{6 x})$

Step 14.3.2

Rewrite $\ln (36 e^{6 x})$ as $\ln (36) + \ln (e^{6 x})$ .

$- (6 x - 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + \ln (e^{6 x}))$

Step 14.3.3

Expand $\ln (e^{6 x})$ by moving $6 x$ outside the logarithm.

$- (6 x - 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x \ln (e))$

Step 14.3.4

The natural logarithm of $e$ is $1$ .

$- (6 x - 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x \cdot 1)$

Step 14.3.5

Multiply $6$ by $1$ .

$- (6 x - 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x)$

Step 14.4

Simplify the left side.

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Step 14.4.1

Simplify $- (6 x - 5 y)$ .

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Step 14.4.1.1

Apply the distributive property.

$- (6 x) - (- 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x)$

Step 14.4.1.2

Multiply.

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Step 14.4.1.2.1

Multiply $6$ by $- 1$ .

$- 6 x - (- 5 y) = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x)$

Step 14.4.1.2.2

Multiply $- 5$ by $- 1$ .

$- 6 x + 5 y = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x)$

Step 14.5

Simplify the right side.

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Step 14.5.1

Simplify $\ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - (\ln (36) + 6 x)$ .

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Step 14.5.1.1

Simplify each term.

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Step 14.5.1.1.1

Apply the distributive property.

$- 6 x + 5 y = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - \ln (36) - (6 x)$

Step 14.5.1.1.2

Multiply $6$ by $- 1$ .

$- 6 x + 5 y = \ln (30 x e^{6 x} - 5 e^{6 x} + 36 C) - \ln (36) - 6 x$

Step 14.5.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$- 6 x + 5 y = \ln (\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36}) - 6 x$

Step 14.5.1.3

Simplify each term.

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Step 14.5.1.3.1

Split the fraction $\frac{30 x e^{6 x} - 5 e^{6 x} + 36 C}{36}$ into two fractions.

$- 6 x + 5 y = \ln (\frac{30 x e^{6 x} - 5 e^{6 x}}{36} + \frac{36 C}{36}) - 6 x$

Step 14.5.1.3.2

Simplify each term.

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Step 14.5.1.3.2.1

Factor $5 e^{6 x}$ out of $30 x e^{6 x} - 5 e^{6 x}$ .

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Step 14.5.1.3.2.1.1

Factor $5 e^{6 x}$ out of $30 x e^{6 x}$ .

$- 6 x + 5 y = \ln (\frac{5 e^{6 x} (6 x) - 5 e^{6 x}}{36} + \frac{36 C}{36}) - 6 x$

Step 14.5.1.3.2.1.2

Factor $5 e^{6 x}$ out of $- 5 e^{6 x}$ .

$- 6 x + 5 y = \ln (\frac{5 e^{6 x} (6 x) + 5 e^{6 x} (- 1)}{36} + \frac{36 C}{36}) - 6 x$

Step 14.5.1.3.2.1.3

Factor $5 e^{6 x}$ out of $5 e^{6 x} (6 x) + 5 e^{6 x} (- 1)$ .

$- 6 x + 5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + \frac{36 C}{36}) - 6 x$

Step 14.5.1.3.2.2

Cancel the common factor of $36$ .

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Step 14.5.1.3.2.2.1

Cancel the common factor.

$- 6 x + 5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + \frac{36 C}{36}) - 6 x$

Step 14.5.1.3.2.2.2

Divide $C$ by $1$ .

$- 6 x + 5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C) - 6 x$

Step 14.6

Move all terms not containing $y$ to the right side of the equation.

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Step 14.6.1

Add $6 x$ to both sides of the equation.

$5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C) - 6 x + 6 x$

Step 14.6.2

Combine the opposite terms in $\ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C) - 6 x + 6 x$ .

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Step 14.6.2.1

Add $- 6 x$ and $6 x$ .

$5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C) + 0$

Step 14.6.2.2

Add $\ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)$ and $0$ .

$5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)$

Step 14.7

Divide each term in $5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)$ by $5$ and simplify.

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Step 14.7.1

Divide each term in $5 y = \ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)$ by $5$ .

$\frac{5 y}{5} = \frac{\ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)}{5}$

Step 14.7.2

Simplify the left side.

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Step 14.7.2.1

Cancel the common factor of $5$ .

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Step 14.7.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{\ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)}{5}$

Step 14.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{5 e^{6 x} (6 x - 1)}{36} + C)}{5}$