Precalculus Examples

$f (x) = 9 x$ , $x = 2$

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.

$f (x + h) = 9 (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 9 x + 9 h$

Step 2.1.2.2

The final answer is $9 x + 9 h$ .

$9 x + 9 h$

Step 2.2

Find the components of the definition.

$f (x + h) = 9 x + 9 h$

$f (x) = 9 x$

$f (x + h) = 9 x + 9 h$

$f (x) = 9 x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{9 x + 9 h - (9 x)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 1$ by $9$ .

$\frac{9 x + 9 h - 9 x}{h}$

Step 4.1.2

Subtract $9 x$ from $9 x$ .

$\frac{9 h + 0}{h}$

Step 4.1.3

Add $9 h$ and $0$ .

$\frac{9 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{9 h}{h}$

Step 4.2.2

Divide $9$ by $1$ .

$9$

Step 5

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