Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Simplify the numerator.

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Step 3.8.2.1

Simplify each term.

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Step 3.8.2.1.1

Expand $(x - y) (1 + y')$ using the FOIL Method.

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Step 3.8.2.1.1.1

Apply the distributive property.

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

Step 3.8.2.1.1.2

Apply the distributive property.

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

Step 3.8.2.1.1.3

Apply the distributive property.

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

Step 3.8.2.1.2

Simplify each term.

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Step 3.8.2.1.2.1

Multiply $x$ by $1$ .

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

Step 3.8.2.1.2.2

Multiply $- 1$ by $1$ .

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

Step 3.8.2.1.3

Expand $(- x - y) (1 - y')$ using the FOIL Method.

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Step 3.8.2.1.3.1

Apply the distributive property.

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

Step 3.8.2.1.3.2

Apply the distributive property.

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

Step 3.8.2.1.3.3

Apply the distributive property.

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

Step 3.8.2.1.4

Simplify each term.

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Step 3.8.2.1.4.1

Multiply $- 1$ by $1$ .

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

Step 3.8.2.1.4.2

Multiply $- x (- y')$ .

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Step 3.8.2.1.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.8.2.1.4.2.2

Multiply $x$ by $1$ .

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

Step 3.8.2.1.4.3

Multiply $- 1$ by $1$ .

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

Step 3.8.2.1.4.4

Multiply $- y (- y')$ .

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Step 3.8.2.1.4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.8.2.1.4.4.2

Multiply $y$ by $1$ .

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

Step 3.8.2.2

Combine the opposite terms in $x + x y' - y - y y' - x + x y' - y + y y'$ .

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Step 3.8.2.2.1

Subtract $x$ from $x$ .

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

Step 3.8.2.2.2

Add $x y' - y - y y'$ and $0$ .

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

Step 3.8.2.2.3

Add $- y y'$ and $y y'$ .

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

Step 3.8.2.2.4

Add $x y' - y + x y' - y$ and $0$ .

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

Step 3.8.2.3

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

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

Step 3.8.2.4

Subtract $y$ from $- y$ .

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

Step 3.8.3

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

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Step 3.8.3.1

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

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

Step 3.8.3.2

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

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

Step 3.8.3.3

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(x - y)}^{2}$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$y' {(x - y)}^{2} = 2 (x y' - y)$

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Multiply $- 1$ by $2$ .

$y' {(x - y)}^{2} = 2 x y' - 2 y$

Step 5.2.1.3.2

Move $x$ .

$y' {(x - y)}^{2} = 2 y' x - 2 y$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.1.1

Subtract $2 y' x$ from both sides of the equation.

$y' {(x - y)}^{2} - 2 y' x = - 2 y$

Step 5.3.1.2

Simplify each term.

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Step 5.3.1.2.1

Rewrite ${(x - y)}^{2}$ as $(x - y) (x - y)$ .

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

Step 5.3.1.2.2

Expand $(x - y) (x - y)$ using the FOIL Method.

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Step 5.3.1.2.2.1

Apply the distributive property.

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

Step 5.3.1.2.2.2

Apply the distributive property.

$y' (x \cdot x + x (- y) - y (x - y)) - 2 y' x = - 2 y$

Step 5.3.1.2.2.3

Apply the distributive property.

$y' (x \cdot x + x (- y) - y x - y (- y)) - 2 y' x = - 2 y$

Step 5.3.1.2.3

Simplify and combine like terms.

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Step 5.3.1.2.3.1

Simplify each term.

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Step 5.3.1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.2.3.1.2

Rewrite using the commutative property of multiplication.

$y' (x^{2} - x y - y x - y (- y)) - 2 y' x = - 2 y$

Step 5.3.1.2.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.2.3.1.4

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

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Step 5.3.1.2.3.1.4.1

Move $y$ .

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

Step 5.3.1.2.3.1.4.2

Multiply $y$ by $y$ .

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

Step 5.3.1.2.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.1.2.3.1.6

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

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

Step 5.3.1.2.3.2

Subtract $y x$ from $- x y$ .

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Step 5.3.1.2.3.2.1

Move $y$ .

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

Step 5.3.1.2.3.2.2

Subtract $x y$ from $- x y$ .

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

Step 5.3.1.2.4

Apply the distributive property.

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

Step 5.3.1.2.5

Rewrite using the commutative property of multiplication.

$y' x^{2} - 2 y' x y + y' y^{2} - 2 y' x = - 2 y$

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.2.4

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

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

Step 5.3.2.5

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

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

Step 5.3.2.6

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

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

Step 5.3.2.7

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

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

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3.3

Simplify the right side.

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Step 5.3.3.3.1

Move the negative in front of the fraction.

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

Step 6

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

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