Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 $f (x) = x y - y^{2}$ and $g (x) = y - x$ .

$\frac{(y - x) \frac{d}{d x} [x y - y^{2}] - (x y - y^{2}) \frac{d}{d x} [y - x]}{{(y - x)}^{2}}$

Step 2.2

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

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

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

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

Step 2.4

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

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

Step 2.5

Differentiate.

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Step 2.5.1

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

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

Step 2.5.2

Multiply $y$ by $1$ .

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

Step 2.5.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

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

Step 2.6

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 2.6.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 2.6.2

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

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

Step 2.6.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.7

Multiply $2$ by $- 1$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

Multiply $- 1$ by $1$ .

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

Step 2.14

Simplify.

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Step 2.14.1

Apply the distributive property.

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

Step 2.14.2

Simplify the numerator.

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Step 2.14.2.1

Simplify each term.

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Step 2.14.2.1.1

Expand $(y - x) (x y' + y - 2 y y')$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2.14.2.1.2

Simplify each term.

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Step 2.14.2.1.2.1

Multiply $y$ by $y$ .

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

Step 2.14.2.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.14.2.1.2.3

Multiply $y$ by $y$ by adding the exponents.

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Step 2.14.2.1.2.3.1

Move $y$ .

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

Step 2.14.2.1.2.3.2

Multiply $y$ by $y$ .

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

Step 2.14.2.1.2.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.14.2.1.2.4.1

Move $x$ .

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

Step 2.14.2.1.2.4.2

Multiply $x$ by $x$ .

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

Step 2.14.2.1.2.5

Multiply $- 2$ by $- 1$ .

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

Step 2.14.2.1.3

Add $y (x y')$ and $2 x y y'$ .

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Step 2.14.2.1.3.1

Reorder $y$ and $x$ .

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

Step 2.14.2.1.3.2

Add $x y y'$ and $2 x y y'$ .

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

Step 2.14.2.1.4

Multiply $- - y^{2}$ .

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Step 2.14.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.14.2.1.4.2

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

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

Step 2.14.2.1.5

Expand $(- x y + y^{2}) (y' - 1)$ using the FOIL Method.

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Step 2.14.2.1.5.1

Apply the distributive property.

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

Step 2.14.2.1.5.2

Apply the distributive property.

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

Step 2.14.2.1.5.3

Apply the distributive property.

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

Step 2.14.2.1.6

Simplify each term.

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Step 2.14.2.1.6.1

Multiply $- x y \cdot - 1$ .

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Step 2.14.2.1.6.1.1

Multiply $- 1$ by $- 1$ .

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

Step 2.14.2.1.6.1.2

Multiply $x$ by $1$ .

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

Step 2.14.2.1.6.2

Move $- 1$ to the left of $y^{2}$ .

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

Step 2.14.2.1.6.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 2.14.2.2

Combine the opposite terms in $y^{2} - 2 y^{2} y' - x^{2} y' - x y + 3 x y y' - x y y' + x y + y^{2} y' - y^{2}$ .

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Step 2.14.2.2.1

Add $- x y$ and $x y$ .

$\frac{y^{2} - 2 y^{2} y' - x^{2} y' + 3 x y y' - x y y' + 0 + y^{2} y' - y^{2}}{{(y - x)}^{2}}$

Step 2.14.2.2.2

Add $y^{2} - 2 y^{2} y' - x^{2} y' + 3 x y y' - x y y'$ and $0$ .

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

Step 2.14.2.2.3

Subtract $y^{2}$ from $y^{2}$ .

$\frac{- 2 y^{2} y' - x^{2} y' + 3 x y y' - x y y' + y^{2} y' + 0}{{(y - x)}^{2}}$

Step 2.14.2.2.4

Add $- 2 y^{2} y' - x^{2} y' + 3 x y y' - x y y' + y^{2} y'$ and $0$ .

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

Step 2.14.2.3

Add $- 2 y^{2} y'$ and $y^{2} y'$ .

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

Step 2.14.2.4

Subtract $x y y'$ from $3 x y y'$ .

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

Step 2.14.3

Reorder terms.

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

Step 2.14.4

Simplify the numerator.

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Step 2.14.4.1

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

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Step 2.14.4.1.1

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

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

Step 2.14.4.1.2

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

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

Step 2.14.4.1.3

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

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

Step 2.14.4.1.4

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

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

Step 2.14.4.1.5

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

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

Step 2.14.4.2

Factor by grouping.

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Step 2.14.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 2.14.4.2.1.1

Reorder terms.

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

Step 2.14.4.2.1.2

Reorder $- y^{2}$ and $2 x y$ .

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

Step 2.14.4.2.1.3

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

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

Step 2.14.4.2.1.4

Rewrite $2$ as $1$ plus $1$

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

Step 2.14.4.2.1.5

Apply the distributive property.

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

Step 2.14.4.2.1.6

Multiply $x y$ by $1$ .

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

Step 2.14.4.2.1.7

Multiply $x y$ by $1$ .

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

Step 2.14.4.2.2

Factor out the greatest common factor from each group.

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Step 2.14.4.2.2.1

Group the first two terms and the last two terms.

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

Step 2.14.4.2.2.2

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

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

Step 2.14.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + y$ .

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

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

Step 2.14.4.3

Combine exponents.

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Step 2.14.4.3.1

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

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

Step 2.14.4.3.2

Factor $- 1$ out of $y$ .

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

Step 2.14.4.3.3

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

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

Step 2.14.4.3.4

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

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

Step 2.14.4.3.5

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

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

Step 2.14.4.3.6

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

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

Step 2.14.4.3.7

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

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

Step 2.14.4.3.8

Add $1$ and $1$ .

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

Step 2.14.4.4

Factor out negative.

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

Step 2.14.5

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

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Step 2.14.5.1

Factor $- 1$ out of $x$ .

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

Step 2.14.5.2

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

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

Step 2.14.5.3

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

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

Step 2.14.5.4

Apply the product rule to $- 1 (- x + y)$ .

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

Step 2.14.5.5

Raise $- 1$ to the power of $2$ .

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

Step 2.14.5.6

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

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

Step 2.14.5.7

Cancel the common factor.

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

Step 2.14.5.8

Divide $- y'$ by $1$ .

$- y'$

Step 3

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

$0$

Step 4

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

$- y' = 0$

Step 5

Divide each term in $- y' = 0$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- y' = 0$ by $- 1$ .

$\frac{- y'}{- 1} = \frac{0}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{y'}{1} = \frac{0}{- 1}$

Step 5.2.2

Divide $y'$ by $1$ .

$y' = \frac{0}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 1$ .

$y' = 0$

Step 6

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

$\frac{d y}{d x} = 0$