Calculus Examples

$\ln (x^{2} + y^{2}) + 2 \arctan (\frac{x}{y}) = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\ln (x^{2} + y^{2}) + 2 \arctan (\frac{x}{y})) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

$\frac{d}{d x} [\ln (x^{2} + y^{2})] + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2

Evaluate $\frac{d}{d x} [\ln (x^{2} + y^{2})]$ .

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Step 2.2.1

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

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Step 2.2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [x^{2} + y^{2}] + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

$\frac{1}{u_{1}} \frac{d}{d x} [x^{2} + y^{2}] + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

$\frac{1}{x^{2} + y^{2}} \frac{d}{d x} [x^{2} + y^{2}] + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.2

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

$\frac{1}{x^{2} + y^{2}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]) + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.3

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

$\frac{1}{x^{2} + y^{2}} (2 x + \frac{d}{d x} [y^{2}]) + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.4

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

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Step 2.2.4.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$\frac{1}{x^{2} + y^{2}} (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.4.2

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

$\frac{1}{x^{2} + y^{2}} (2 x + 2 u_{2} \frac{d}{d x} [y]) + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.4.3

Replace all occurrences of $u_{2}$ with $y$ .

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y \frac{d}{d x} [y]) + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.2.5

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

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + \frac{d}{d x} [2 \arctan (\frac{x}{y})]$

Step 2.3

Evaluate $\frac{d}{d x} [2 \arctan (\frac{x}{y})]$ .

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Step 2.3.1

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

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 \frac{d}{d x} [\arctan (\frac{x}{y})]$

Step 2.3.2

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

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Step 2.3.2.1

To apply the Chain Rule, set $u_{3}$ as $\frac{x}{y}$ .

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 (\frac{d}{d u_{3}} [\arctan (u_{3})] \frac{d}{d x} [\frac{x}{y}])$

Step 2.3.2.2

The derivative of $\arctan (u_{3})$ with respect to $u_{3}$ is $\frac{1}{1 + {u_{3}}^{2}}$ .

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 (\frac{1}{1 + {u_{3}}^{2}} \frac{d}{d x} [\frac{x}{y}])$

Step 2.3.2.3

Replace all occurrences of $u_{3}$ with $\frac{x}{y}$ .

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 (\frac{1}{1 + {(\frac{x}{y})}^{2}} \frac{d}{d x} [\frac{x}{y}])$

Step 2.3.3

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 $f (x) = x$ and $g (x) = y$ .

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 (\frac{1}{1 + {(\frac{x}{y})}^{2}} \cdot \frac{y \frac{d}{d x} [x] - x \frac{d}{d x} [y]}{y^{2}})$

Step 2.3.4

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

$\frac{1}{x^{2} + y^{2}} (2 x + 2 y y') + 2 (\frac{1}{1 + {(\frac{x}{y})}^{2}} \cdot \frac{y \cdot 1 - x \frac{d}{d x} [y]}{y^{2}})$

Step 2.3.5

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

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

Step 2.3.6

Multiply $y$ by $1$ .

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

Step 2.3.7

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

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

Step 2.3.8

Combine $2$ and $\frac{y - x y'}{(1 + {(\frac{x}{y})}^{2}) y^{2}}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the product rule to $\frac{x}{y}$ .

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.4.4

Combine terms.

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Step 2.4.4.1

Multiply $- 1$ by $2$ .

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

Step 2.4.4.2

Multiply $y^{2}$ by $1$ .

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

Step 2.4.4.3

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

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

Step 2.4.4.4

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

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Step 2.4.4.4.1

Cancel the common factor.

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

Step 2.4.4.4.2

Divide $x^{2}$ by $1$ .

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

Step 2.4.4.5

To write $\frac{1}{x^{2} + y^{2}} (2 x + 2 y y')$ as a fraction with a common denominator, multiply by $\frac{y^{2} + x^{2}}{y^{2} + x^{2}}$ .

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

Step 2.4.4.6

Combine $\frac{1}{x^{2} + y^{2}} (2 x + 2 y y')$ and $\frac{y^{2} + x^{2}}{y^{2} + x^{2}}$ .

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

Step 2.4.4.7

Combine the numerators over the common denominator.

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

Step 2.4.5

Reorder terms.

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

Step 2.4.6

Simplify the numerator.

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Step 2.4.6.1

Expand $(2 x + 2 y y') (y^{2} + x^{2})$ using the FOIL Method.

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Step 2.4.6.1.1

Apply the distributive property.

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

Step 2.4.6.1.2

Apply the distributive property.

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

Step 2.4.6.1.3

Apply the distributive property.

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

Step 2.4.6.2

Simplify each term.

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Step 2.4.6.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.4.6.2.1.1

Move $x^{2}$ .

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

Step 2.4.6.2.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.4.6.2.1.2.1

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

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

Step 2.4.6.2.1.2.2

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

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

Step 2.4.6.2.1.3

Add $2$ and $1$ .

$\frac{(2 x y^{2} + 2 x^{3} + 2 y y' y^{2} + 2 y y' x^{2}) \frac{1}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.2.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.4.6.2.2.1

Move $y^{2}$ .

$\frac{(2 x y^{2} + 2 x^{3} + 2 (y^{2} y) y' + 2 y y' x^{2}) \frac{1}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.2.2.2

Multiply $y^{2}$ by $y$ .

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Step 2.4.6.2.2.2.1

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

$\frac{(2 x y^{2} + 2 x^{3} + 2 (y^{2} y^{1}) y' + 2 y y' x^{2}) \frac{1}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.2.2.2.2

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

$\frac{(2 x y^{2} + 2 x^{3} + 2 y^{2 + 1} y' + 2 y y' x^{2}) \frac{1}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.2.2.3

Add $2$ and $1$ .

$\frac{(2 x y^{2} + 2 x^{3} + 2 y^{3} y' + 2 y y' x^{2}) \frac{1}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.3

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

$\frac{\frac{2 x y^{2} + 2 x^{3} + 2 y^{3} y' + 2 y y' x^{2}}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4

Simplify the numerator.

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Step 2.4.6.4.1

Factor $2$ out of $2 x y^{2} + 2 x^{3} + 2 y^{3} y' + 2 y y' x^{2}$ .

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Step 2.4.6.4.1.1

Factor $2$ out of $2 x y^{2}$ .

$\frac{\frac{2 (x y^{2}) + 2 x^{3} + 2 y^{3} y' + 2 y y' x^{2}}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.2

Factor $2$ out of $2 x^{3}$ .

$\frac{\frac{2 (x y^{2}) + 2 (x^{3}) + 2 y^{3} y' + 2 y y' x^{2}}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.3

Factor $2$ out of $2 y^{3} y'$ .

$\frac{\frac{2 (x y^{2}) + 2 (x^{3}) + 2 (y^{3} y') + 2 y y' x^{2}}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.4

Factor $2$ out of $2 y y' x^{2}$ .

$\frac{\frac{2 (x y^{2}) + 2 (x^{3}) + 2 (y^{3} y') + 2 (y y' x^{2})}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.5

Factor $2$ out of $2 (x y^{2}) + 2 (x^{3})$ .

$\frac{\frac{2 (x y^{2} + x^{3}) + 2 (y^{3} y') + 2 (y y' x^{2})}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.6

Factor $2$ out of $2 (x y^{2} + x^{3}) + 2 (y^{3} y')$ .

$\frac{\frac{2 (x y^{2} + x^{3} + y^{3} y') + 2 (y y' x^{2})}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.1.7

Factor $2$ out of $2 (x y^{2} + x^{3} + y^{3} y') + 2 (y y' x^{2})$ .

$\frac{\frac{2 (x y^{2} + x^{3} + y^{3} y' + y y' x^{2})}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.2

Factor out the greatest common factor from each group.

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Step 2.4.6.4.2.1

Group the first two terms and the last two terms.

$\frac{\frac{2 ((x y^{2} + x^{3}) + y^{3} y' + y y' x^{2})}{x^{2} + y^{2}} - 2 x y' + 2 y}{x^{2} + y^{2}}$

Step 2.4.6.4.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.4.6.4.3

Factor the polynomial by factoring out the greatest common factor, $y^{2} + x^{2}$ .

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

Step 2.4.6.5

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

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Step 2.4.6.5.1

Reorder terms.

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

Step 2.4.6.5.2

Cancel the common factor.

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

Step 2.4.6.5.3

Divide $2 (x + y y')$ by $1$ .

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

Step 2.4.6.6

Apply the distributive property.

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

Step 2.4.6.7

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

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Step 2.4.6.7.1

Factor $2$ out of $2 x$ .

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

Step 2.4.6.7.2

Factor $2$ out of $2 y y'$ .

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

Step 2.4.6.7.3

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

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

Step 2.4.6.7.4

Factor $2$ out of $2 y$ .

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

Step 2.4.6.7.5

Factor $2$ out of $2 (x) + 2 (y y')$ .

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

Step 2.4.6.7.6

Factor $2$ out of $2 (x + y y') + 2 (- x y')$ .

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

Step 2.4.6.7.7

Factor $2$ out of $2 (x + y y' - x y') + 2 (y)$ .

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

Step 3

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

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$\frac{2 (x + y y' - x y' + y)}{x^{2} + y^{2}} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$2 (x + y y' - x y' + y) = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Divide each term in $2 (x + y y' - x y' + y) = 0$ by $2$ and simplify.

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Step 5.2.1.1

Divide each term in $2 (x + y y' - x y' + y) = 0$ by $2$ .

$\frac{2 (x + y y' - x y' + y)}{2} = \frac{0}{2}$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x + y y' - x y' + y)}{2} = \frac{0}{2}$

Step 5.2.1.2.1.2

Divide $x + y y' - x y' + y$ by $1$ .

$x + y y' - x y' + y = \frac{0}{2}$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Divide $0$ by $2$ .

$x + y y' - x y' + y = 0$

Step 5.2.2

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

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Step 5.2.2.1

Subtract $x$ from both sides of the equation.

$y y' - x y' + y = - x$

Step 5.2.2.2

Subtract $y$ from both sides of the equation.

$y y' - x y' = - x - y$

Step 5.2.3

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

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Step 5.2.3.1

Factor $y'$ out of $y y'$ .

$y' y - x y' = - x - y$

Step 5.2.3.2

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

$y' (y) + y' (- x) = - x - y$

Step 5.2.3.3

Factor $y'$ out of $y' (y) + y' (- x)$ .

$y' (y - x) = - x - y$

Step 5.2.4

Divide each term in $y' (y - x) = - x - y$ by $y - x$ and simplify.

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Step 5.2.4.1

Divide each term in $y' (y - x) = - x - y$ by $y - x$ .

$\frac{y' (y - x)}{y - x} = \frac{- x}{y - x} + \frac{- y}{y - x}$

Step 5.2.4.2

Simplify the left side.

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Step 5.2.4.2.1

Cancel the common factor of $y - x$ .

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Step 5.2.4.2.1.1

Cancel the common factor.

$\frac{y' (y - x)}{y - x} = \frac{- x}{y - x} + \frac{- y}{y - x}$

Step 5.2.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- x}{y - x} + \frac{- y}{y - x}$

Step 5.2.4.3

Simplify the right side.

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Step 5.2.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{- x - y}{y - x}$

Step 5.2.4.3.2

Factor $- 1$ out of $- x$ .

$y' = \frac{- (x) - y}{y - x}$

Step 5.2.4.3.3

Factor $- 1$ out of $- y$ .

$y' = \frac{- (x) - (y)}{y - x}$

Step 5.2.4.3.4

Factor $- 1$ out of $- (x) - (y)$ .

$y' = \frac{- (x + y)}{y - x}$

Step 5.2.4.3.5

Simplify the expression.

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Step 5.2.4.3.5.1

Rewrite $- (x + y)$ as $- 1 (x + y)$ .

$y' = \frac{- 1 (x + y)}{y - x}$

Step 5.2.4.3.5.2

Move the negative in front of the fraction.

$y' = - \frac{x + y}{y - x}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{x + y}{y - x}$