Calculus Examples

$f (x) = \frac{1}{x - 1}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{1}{(x + h) - 1}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{1}{x + h - 1}$

Step 2.1.2.2

The final answer is $\frac{1}{x + h - 1}$ .

$\frac{1}{x + h - 1}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{1}{x + h - 1}$

$f (x) = \frac{1}{x - 1}$

$f (x + h) = \frac{1}{x + h - 1}$

$f (x) = \frac{1}{x - 1}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1}{x + h - 1} - (\frac{1}{x - 1})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{x + h - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{\frac{1}{x + h - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1}}{h}$

Step 4.1.2

To write $- \frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

$\frac{\frac{1}{x + h - 1} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{x + h - 1}{x + h - 1}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h - 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{1}{x + h - 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{1}{x - 1} \cdot \frac{x + h - 1}{x + h - 1}}{h}$

Step 4.1.3.2

Multiply $\frac{1}{x - 1}$ by $\frac{x + h - 1}{x + h - 1}$ .

$\frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{x + h - 1}{(x - 1) (x + h - 1)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x - 1) (x + h - 1)$ .

$\frac{\frac{x - 1}{(x + h - 1) (x - 1)} - \frac{x + h - 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{x - 1 - (x + h - 1)}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5

Rewrite $\frac{x - 1 - (x + h - 1)}{(x + h - 1) (x - 1)}$ in a factored form.

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Step 4.1.5.1

Apply the distributive property.

$\frac{\frac{x - 1 - x - h - - 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.2

Multiply $- 1$ by $- 1$ .

$\frac{\frac{x - 1 - x - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.3

Subtract $x$ from $x$ .

$\frac{\frac{0 - 1 - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.4

Subtract $1$ from $0$ .

$\frac{\frac{- 1 - h + 1}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.5

Add $- 1$ and $1$ .

$\frac{\frac{- h + 0}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.6

Add $- h$ and $0$ .

$\frac{\frac{- h}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{h}{(x + h - 1) (x - 1)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{h}{(x + h - 1) (x - 1)}$ into the numerator.

$\frac{- h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 1}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 1}{(x + h - 1) (x - 1)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1}{(x + h - 1) (x - 1)}$

Step 5