Calculus Examples

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

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 $f (x) = 1 - x$ and $g (x) = x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $1 - x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

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

Step 2.6.2

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

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

Step 2.6.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.7

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

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

Step 2.8

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{- x - 1 \cdot 1 - - x}{x^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 1$ by $1$ .

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

Step 3.2.1.2

Multiply $- - x$ .

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Step 3.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- x - 1 + 1 x}{x^{2}}$

Step 3.2.1.2.2

Multiply $x$ by $1$ .

$\frac{- x - 1 + x}{x^{2}}$

Step 3.2.2

Combine the opposite terms in $- x - 1 + x$ .

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Step 3.2.2.1

Add $- x$ and $x$ .

$\frac{0 - 1}{x^{2}}$

Step 3.2.2.2

Subtract $1$ from $0$ .

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

Step 3.3

Move the negative in front of the fraction.

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