Precalculus Examples

$c (x) = 2^{x} - 1$

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.

$c (x + h) = 2^{x + h} - 1$

Step 2.1.2

The final answer is $2^{x + h} - 1$ .

$2^{x + h} - 1$

Step 2.2

Find the components of the definition.

$c (x + h) = 2^{x + h} - 1$

$c (x) = 2^{x} - 1$

$c (x + h) = 2^{x + h} - 1$

$c (x) = 2^{x} - 1$

Step 3

Plug in the components.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3

Add $- 1$ and $1$ .

$\frac{2^{x + h} + 0 - 2^{x}}{h}$

Step 4.4

Add $2^{x + h}$ and $0$ .

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

Step 5