Calculus Examples

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

Step 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 2

Differentiate.

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Step 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 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 2.3

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

$\frac{(x - y) (1 + 0) - (x + y) \frac{d}{d x} [x - y]}{{(x - y)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $x - y$ by $1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

$\frac{x - y - (x + y) (1 + 0)}{{(x - y)}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

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

Step 2.8.2

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

Simplify the numerator.

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Step 3.2.1

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

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Step 3.2.1.1

Subtract $x$ from $x$ .

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

Step 3.2.1.2

Add $- y$ and $0$ .

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

Step 3.2.2

Subtract $y$ from $- y$ .

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

Step 3.3

Move the negative in front of the fraction.

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