Precalculus Examples

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

Step 1

Write $\frac{f (x) - f (1)}{x - 1}$ as a function.

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

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Factor $f$ out of $f (x + h) - f \cdot 1$ .

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Step 3.1.2.1.1.1

Factor $f$ out of $- f \cdot 1$ .

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

Step 3.1.2.1.1.2

Factor $f$ out of $f (x + h) + f (- 1 \cdot 1)$ .

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

Step 3.1.2.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.2.2

Cancel the common factor of $x + h - 1$ .

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Step 3.1.2.2.1

Cancel the common factor.

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

Step 3.1.2.2.2

Divide $f$ by $1$ .

$f (x + h) = f$

Step 3.1.2.3

The final answer is $f$ .

$f$

Step 3.2

Find the components of the definition.

$f (x + h) = f$

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

$f (x + h) = f$

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

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Combine the numerators over the common denominator.

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

Step 5.1.3

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

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Step 5.1.3.1

Apply the distributive property.

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Rewrite $- 1 f$ as $- f$ .

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

Step 5.1.3.4

Apply the distributive property.

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

Step 5.1.3.5

Multiply $- - f$ .

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Step 5.1.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.3.5.2

Multiply $f$ by $1$ .

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

Step 5.1.3.6

Add $- f$ and $f$ .

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

Step 5.1.3.7

Add $f x$ and $0$ .

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Multiply $\frac{f x - f (x)}{x - 1}$ by $\frac{1}{h}$ .

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

Step 5.4

Reorder factors in $\frac{f x - f (x)}{(x - 1) h}$ .

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

Step 6