Precalculus Examples

$\frac{f (b) - f (a)}{b - a}$

Step 1

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

$f (b) = \frac{f (b) - f (a)}{b - a}$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

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

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 $f (b + h) - f a$ .

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Step 3.1.2.1.1.1

Factor $f$ out of $- f a$ .

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

Step 3.1.2.1.1.2

Factor $f$ out of $f (b + h) + f (- 1 a)$ .

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

Step 3.1.2.1.2

Rewrite $- 1 a$ as $- a$ .

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

Step 3.1.2.2

Cancel the common factor of $b + h - a$ .

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Step 3.1.2.2.1

Cancel the common factor.

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

Step 3.1.2.2.2

Divide $f$ by $1$ .

$f (b + h) = f$

Step 3.1.2.3

The final answer is $f$ .

$f$

Step 3.2

Find the components of the definition.

$f (b + h) = f$

$f (b) = f$

$f (b + h) = f$

$f (b) = f$

Step 4

Plug in the components.

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

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