Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Flip sides to get $\frac{d x}{d y}$ on the left side.

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

Step 1.2

Divide each term in $9 (x - 2 x y) \frac{d x}{d y} = 9$ by $9 (x - 2 x y)$ and simplify.

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Step 1.2.1

Divide each term in $9 (x - 2 x y) \frac{d x}{d y} = 9$ by $9 (x - 2 x y)$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Rewrite the expression.

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

Step 1.2.2.2

Cancel the common factor of $x - 2 x y$ .

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Step 1.2.2.2.1

Cancel the common factor.

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

Step 1.2.2.2.2

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

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $9$ .

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Step 1.2.3.1.1

Cancel the common factor.

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

Step 1.2.3.1.2

Rewrite the expression.

$\frac{d x}{d y} = \frac{1}{x - 2 x y}$

Step 1.2.3.2

Factor $x$ out of $x - 2 x y$ .

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

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

$\frac{d x}{d y} = \frac{1}{x \cdot 1 - 2 x y}$

Step 1.2.3.2.3

Factor $x$ out of $- 2 x y$ .

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

Step 1.2.3.2.4

Factor $x$ out of $x \cdot 1 + x (- 2 y)$ .

$\frac{d x}{d y} = \frac{1}{x (1 - 2 y)}$

Step 1.3

Regroup factors.

$\frac{d x}{d y} = \frac{1}{x} \cdot \frac{1}{1 - 2 y}$

Step 1.4

Multiply both sides by $x$ .

$x \frac{d x}{d y} = x (\frac{1}{x} \cdot \frac{1}{1 - 2 y})$

Step 1.5

Simplify.

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Step 1.5.1

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

$x \frac{d x}{d y} = x \frac{1}{x (1 - 2 y)}$

Step 1.5.2

Cancel the common factor of $x$ .

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Step 1.5.2.1

Cancel the common factor.

$x \frac{d x}{d y} = x \frac{1}{x (1 - 2 y)}$

Step 1.5.2.2

Rewrite the expression.

$x \frac{d x}{d y} = \frac{1}{1 - 2 y}$

Step 1.6

Rewrite the equation.

$x d x = \frac{1}{1 - 2 y} d y$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int x d x = \int \frac{1}{1 - 2 y} d y$

Step 2.2

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

$\frac{1}{2} x^{2} + C_{1} = \int \frac{1}{1 - 2 y} d y$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 1 - 2 y$ . Then $d u = - 2 d y$ , so $- \frac{1}{2} d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 1 - 2 y$ . Find $\frac{d u}{d y}$ .

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Step 2.3.1.1.1

Differentiate $1 - 2 y$ .

$\frac{d}{d y} [1 - 2 y]$

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

By the Sum Rule, the derivative of $1 - 2 y$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- 2 y]$ .

$\frac{d}{d y} [1] + \frac{d}{d y} [- 2 y]$

Step 2.3.1.1.2.2

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

$0 + \frac{d}{d y} [- 2 y]$

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

$0 - 2 \frac{d}{d y} [y]$

Step 2.3.1.1.3.2

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

$0 - 2 \cdot 1$

Step 2.3.1.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.3.1.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.3.1.2

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

$\frac{1}{2} x^{2} + C_{1} = \int \frac{1}{u} \cdot \frac{1}{- 2} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$\frac{1}{2} x^{2} + C_{1} = \int \frac{1}{u} (- \frac{1}{2}) d u$

Step 2.3.2.2

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$\frac{1}{2} x^{2} + C_{1} = \int - \frac{1}{u \cdot 2} d u$

Step 2.3.2.3

Move $2$ to the left of $u$ .

$\frac{1}{2} x^{2} + C_{1} = \int - \frac{1}{2 u} d u$

Step 2.3.3

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

$\frac{1}{2} x^{2} + C_{1} = - \int \frac{1}{2 u} d u$

Step 2.3.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} x^{2} + C_{1} = - (\frac{1}{2} \int \frac{1}{u} d u)$

Step 2.3.5

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

$\frac{1}{2} x^{2} + C_{1} = - \frac{1}{2} (\ln (| u |) + C_{2})$

Step 2.3.6

Simplify.

$\frac{1}{2} x^{2} + C_{1} = - \frac{1}{2} \ln (| u |) + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $1 - 2 y$ .

$\frac{1}{2} x^{2} + C_{1} = - \frac{1}{2} \ln (| 1 - 2 y |) + C_{2}$

Step 2.4

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

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

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} x^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \frac{1}{2} \ln (| 1 - 2 y |) + C)$ .

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Step 3.2.2.1.1

Combine $\ln (| 1 - 2 y |)$ and $\frac{1}{2}$ .

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

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

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

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

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

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Cancel the common factor.

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

Step 3.2.2.1.3.3.2

Rewrite the expression.

$x^{2} = - \ln (| 1 - 2 y |) + C \cdot 2$

Step 3.2.2.1.4

Move $2$ to the left of $C$ .

$x^{2} = - \ln (| 1 - 2 y |) + 2 C$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

$x = - \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

$x = \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

$x = - \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

$x = \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

$x = - \sqrt{- \ln (| 1 - 2 y |) + 2 C}$

Step 4

Simplify the constant of integration.

$x = \sqrt{- \ln (| 1 - 2 y |) + C}$

$x = - \sqrt{- \ln (| 1 - 2 y |) + C}$