Calculus Examples

$\frac{d y}{d x} = 6 \cdot \frac{1}{x + 6} \cdot y$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{y}$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{1}{y} (6 \cdot \frac{1}{x + 6} \cdot y)$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{y} \frac{d y}{d x} = 6 \frac{1}{y} (\frac{1}{x + 6} y)$

Step 1.2.2

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

$\frac{1}{y} \frac{d y}{d x} = \frac{6}{y} (\frac{1}{x + 6} y)$

Step 1.2.3

Cancel the common factor of $y$ .

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Step 1.2.3.1

Factor $y$ out of $\frac{1}{x + 6} y$ .

$\frac{1}{y} \frac{d y}{d x} = \frac{6}{y} (y \frac{1}{x + 6})$

Step 1.2.3.2

Cancel the common factor.

$\frac{1}{y} \frac{d y}{d x} = \frac{6}{y} (y \frac{1}{x + 6})$

Step 1.2.3.3

Rewrite the expression.

$\frac{1}{y} \frac{d y}{d x} = 6 \frac{1}{x + 6}$

Step 1.2.4

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

$\frac{1}{y} \frac{d y}{d x} = \frac{6}{x + 6}$

Step 1.3

Rewrite the equation.

$\frac{1}{y} d y = \frac{6}{x + 6} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{y} d y = \int \frac{6}{x + 6} d x$

Step 2.2

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

$\ln (| y |) + C_{1} = \int \frac{6}{x + 6} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\ln (| y |) + C_{1} = 6 \int \frac{1}{x + 6} d x$

Step 2.3.2

Let $u = x + 6$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $x + 6$ .

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

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

$1 + \frac{d}{d x} [6]$

Step 2.3.2.1.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

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

$\ln (| y |) + C_{1} = 6 \int \frac{1}{u} d u$

Step 2.3.3

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

$\ln (| y |) + C_{1} = 6 (\ln (| u |) + C_{2})$

Step 2.3.4

Simplify.

$\ln (| y |) + C_{1} = 6 \ln (| u |) + C_{2}$

Step 2.3.5

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

$\ln (| y |) + C_{1} = 6 \ln (| x + 6 |) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (| y |) = 6 \ln (| x + 6 |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Move all the terms containing a logarithm to the left side of the equation.

$\ln (| y |) - 6 \ln (| x + 6 |) = C$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify $\ln (| y |) - 6 \ln (| x + 6 |)$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Simplify $- 6 \ln (| x + 6 |)$ by moving $6$ inside the logarithm.

$\ln (| y |) - \ln ({| x + 6 |}^{6}) = C$

Step 3.2.1.1.2

Remove the absolute value in ${| x + 6 |}^{6}$ because exponentiations with even powers are always positive.

$\ln (| y |) - \ln ({(x + 6)}^{6}) = C$

Step 3.2.1.2

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

$\ln (\frac{| y |}{{(x + 6)}^{6}}) = C$

Step 3.3

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Step 3.4

Rewrite $\ln (\frac{| y |}{{(x + 6)}^{6}}) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = \frac{| y |}{{(x + 6)}^{6}}$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $\frac{| y |}{{(x + 6)}^{6}} = e^{C}$ .

$\frac{| y |}{{(x + 6)}^{6}} = e^{C}$

Step 3.5.2

Multiply both sides by ${(x + 6)}^{6}$ .

$\frac{| y |}{{(x + 6)}^{6}} {(x + 6)}^{6} = e^{C} {(x + 6)}^{6}$

Step 3.5.3

Simplify the left side.

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Step 3.5.3.1

Cancel the common factor of ${(x + 6)}^{6}$ .

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Step 3.5.3.1.1

Cancel the common factor.

$\frac{| y |}{{(x + 6)}^{6}} {(x + 6)}^{6} = e^{C} {(x + 6)}^{6}$

Step 3.5.3.1.2

Rewrite the expression.

$| y | = e^{C} {(x + 6)}^{6}$

Step 3.5.4

Solve for $y$ .

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Step 3.5.4.1

Simplify $e^{C} {(x + 6)}^{6}$ .

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Step 3.5.4.1.1

Use the Binomial Theorem.

$| y | = e^{C} (x^{6} + 6 x^{5} \cdot 6 + 15 x^{4} \cdot 6^{2} + 20 x^{3} \cdot 6^{3} + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2

Simplify terms.

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Step 3.5.4.1.2.1

Simplify each term.

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Step 3.5.4.1.2.1.1

Multiply $6$ by $6$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 15 x^{4} \cdot 6^{2} + 20 x^{3} \cdot 6^{3} + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.2

Raise $6$ to the power of $2$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 15 x^{4} \cdot 36 + 20 x^{3} \cdot 6^{3} + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.3

Multiply $36$ by $15$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 20 x^{3} \cdot 6^{3} + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.4

Raise $6$ to the power of $3$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 20 x^{3} \cdot 216 + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.5

Multiply $216$ by $20$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 15 x^{2} \cdot 6^{4} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.6

Raise $6$ to the power of $4$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 15 x^{2} \cdot 1296 + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.7

Multiply $1296$ by $15$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 6 x \cdot 6^{5} + 6^{6})$

Step 3.5.4.1.2.1.8

Multiply $6$ by $6^{5}$ by adding the exponents.

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Step 3.5.4.1.2.1.8.1

Move $6^{5}$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 6^{5} \cdot 6 x + 6^{6})$

Step 3.5.4.1.2.1.8.2

Multiply $6^{5}$ by $6$ .

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Step 3.5.4.1.2.1.8.2.1

Raise $6$ to the power of $1$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 6^{5} \cdot 6^{1} x + 6^{6})$

Step 3.5.4.1.2.1.8.2.2

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

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 6^{5 + 1} x + 6^{6})$

Step 3.5.4.1.2.1.8.3

Add $5$ and $1$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 6^{6} x + 6^{6})$

Step 3.5.4.1.2.1.9

Raise $6$ to the power of $6$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 46656 x + 6^{6})$

Step 3.5.4.1.2.1.10

Raise $6$ to the power of $6$ .

$| y | = e^{C} (x^{6} + 36 x^{5} + 540 x^{4} + 4320 x^{3} + 19440 x^{2} + 46656 x + 46656)$

Step 3.5.4.1.2.2

Apply the distributive property.

$| y | = e^{C} x^{6} + e^{C} (36 x^{5}) + e^{C} (540 x^{4}) + e^{C} (4320 x^{3}) + e^{C} (19440 x^{2}) + e^{C} (46656 x) + e^{C} \cdot 46656$

Step 3.5.4.1.3

Simplify.

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Step 3.5.4.1.3.1

Rewrite using the commutative property of multiplication.

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + e^{C} (540 x^{4}) + e^{C} (4320 x^{3}) + e^{C} (19440 x^{2}) + e^{C} (46656 x) + e^{C} \cdot 46656$

Step 3.5.4.1.3.2

Rewrite using the commutative property of multiplication.

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + e^{C} (4320 x^{3}) + e^{C} (19440 x^{2}) + e^{C} (46656 x) + e^{C} \cdot 46656$

Step 3.5.4.1.3.3

Rewrite using the commutative property of multiplication.

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + 4320 e^{C} x^{3} + e^{C} (19440 x^{2}) + e^{C} (46656 x) + e^{C} \cdot 46656$

Step 3.5.4.1.3.4

Rewrite using the commutative property of multiplication.

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + 4320 e^{C} x^{3} + 19440 e^{C} x^{2} + e^{C} (46656 x) + e^{C} \cdot 46656$

Step 3.5.4.1.3.5

Rewrite using the commutative property of multiplication.

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + 4320 e^{C} x^{3} + 19440 e^{C} x^{2} + 46656 e^{C} x + e^{C} \cdot 46656$

Step 3.5.4.1.3.6

Move $46656$ to the left of $e^{C}$ .

$| y | = e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + 4320 e^{C} x^{3} + 19440 e^{C} x^{2} + 46656 e^{C} x + 46656 \cdot e^{C}$

Step 3.5.4.1.4

Reorder factors in $e^{C} x^{6} + 36 e^{C} x^{5} + 540 e^{C} x^{4} + 4320 e^{C} x^{3} + 19440 e^{C} x^{2} + 46656 e^{C} x + 46656 e^{C}$ .

$| y | = x^{6} e^{C} + 36 x^{5} e^{C} + 540 x^{4} e^{C} + 4320 x^{3} e^{C} + 19440 x^{2} e^{C} + 46656 x e^{C} + 46656 e^{C}$

Step 3.5.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y = \pm (x^{6} e^{C} + 36 x^{5} e^{C} + 540 x^{4} e^{C} + 4320 x^{3} e^{C} + 19440 x^{2} e^{C} + 46656 x e^{C} + 46656 e^{C})$

Step 4

Simplify the constant of integration.

$y = \pm (x^{6} C + 36 x^{5} C + 540 x^{4} C + 4320 x^{3} C + 19440 x^{2} C + 46656 x C + C)$