Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x - y$ and $g (y) = x + y$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

$\frac{(x + y) (0 + \frac{d}{d y} [- y]) - (x - y) \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d y} [- y]$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

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

Step 2.6.2

Move $- 1$ to the left of $x + y$ .

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

Step 2.6.3

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

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

Step 2.7

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

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

Step 2.8

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

$\frac{- (x + y) - (x - y) (0 + \frac{d}{d y} [y])}{{(x + y)}^{2}}$

Step 2.9

Add $0$ and $\frac{d}{d y} [y]$ .

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

Step 2.10

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

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

Step 2.11

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Multiply $- - y$ .

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Step 3.3.1.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.2

Multiply $y$ by $1$ .

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

Step 3.3.2

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

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Step 3.3.2.1

Add $- y$ and $y$ .

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

Step 3.3.2.2

Add $- x - x$ and $0$ .

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

Step 3.3.3

Subtract $x$ from $- x$ .

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

Step 3.4

Move the negative in front of the fraction.

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