Precalculus Examples

$\frac{f (2) - f (1)}{2 - 1}$

Step 1

Write $\frac{f (2) - f (1)}{2 - 1}$ as a function.

$f (x) = \frac{f (2) - f (1)}{2 - 1}$

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.

$f (x + h) = \frac{f (2) - f \cdot 1}{2 - 1}$

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 $f (2) - f \cdot 1$ .

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Step 3.1.2.1.1.1

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

$f (x + h) = \frac{f (2) + f (- 1 \cdot 1)}{2 - 1}$

Step 3.1.2.1.1.2

Factor $f$ out of $f (2) + f (- 1 \cdot 1)$ .

$f (x + h) = \frac{f (2 - 1 \cdot 1)}{2 - 1}$

Step 3.1.2.1.2

Multiply $- 1$ by $1$ .

$f (x + h) = \frac{f (2 - 1)}{2 - 1}$

Step 3.1.2.1.3

Subtract $1$ from $2$ .

$f (x + h) = \frac{f \cdot 1}{2 - 1}$

Step 3.1.2.2

Simplify the expression.

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Step 3.1.2.2.1

Subtract $1$ from $2$ .

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

Step 3.1.2.2.2

Multiply $f$ by $1$ .

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

Step 3.1.2.2.3

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.

$f (x + h) = f$

$f (x) = f$

$f (x + h) = f$

$f (x) = f$

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Subtract $f$ from $f$ .

$\frac{0}{h}$

Step 5.2

Divide $0$ by $h$ .

$0$

Step 6