Precalculus Examples

$f (x) = \frac{1}{2} {(3)}^{x}$

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.

$f (x + h) = \frac{1}{2} \cdot {(3)}^{x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Combine $\frac{1}{2}$ and $3^{x + h}$ .

$f (x + h) = \frac{3^{x + h}}{2}$

Step 2.1.2.2

The final answer is $\frac{3^{x + h}}{2}$ .

$\frac{3^{x + h}}{2}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{3^{x + h}}{2}$

$f (x) = \frac{3^{x}}{2}$

$f (x + h) = \frac{3^{x + h}}{2}$

$f (x) = \frac{3^{x}}{2}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{3^{x + h}}{2} - (\frac{3^{x}}{2})}{h}$

Step 4

Simplify.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{\frac{3^{x + h} - 3^{x}}{2}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3^{x + h} - 3^{x}}{2} \cdot \frac{1}{h}$

Step 4.3

Multiply $\frac{3^{x + h} - 3^{x}}{2}$ by $\frac{1}{h}$ .

$\frac{3^{x + h} - 3^{x}}{2 h}$

Step 5