Precalculus Examples

$N (x) = 90 {(0.86)}^{x} + 69$

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.

$N (x + h) = 90 {(0.86)}^{x + h} + 69$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$N (x + h) = 90 \cdot {0.86}^{x + h} + 69$

Step 2.1.2.2

The final answer is $90 \cdot {0.86}^{x + h} + 69$ .

$90 \cdot {0.86}^{x + h} + 69$

Step 2.2

Reorder $x$ and $h$ .

$90 \cdot {0.86}^{h + x} + 69$

Step 2.3

Find the components of the definition.

$N (x + h) = 90 \cdot {0.86}^{h + x} + 69$

$N (x) = 90 \cdot {0.86}^{x} + 69$

$N (x + h) = 90 \cdot {0.86}^{h + x} + 69$

$N (x) = 90 \cdot {0.86}^{x} + 69$

Step 3

Plug in the components.

$\frac{N (x + h) - N x}{h} = \frac{90 \cdot {0.86}^{h + x} + 69 - (90 \cdot {0.86}^{x} + 69)}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{90 \cdot {0.86}^{h + x} + 69 - (90 \cdot {0.86}^{x}) - 1 \cdot 69}{h}$

Step 4.2

Multiply $90$ by $- 1$ .

$\frac{90 \cdot {0.86}^{h + x} + 69 - 90 \cdot {0.86}^{x} - 1 \cdot 69}{h}$

Step 4.3

Multiply $- 1$ by $69$ .

$\frac{90 \cdot {0.86}^{h + x} + 69 - 90 \cdot {0.86}^{x} - 69}{h}$

Step 4.4

Subtract $69$ from $69$ .

$\frac{90 \cdot {0.86}^{h + x} + 0 - 90 \cdot {0.86}^{x}}{h}$

Step 4.5

Add $90 \cdot {0.86}^{h + x}$ and $0$ .

$\frac{90 \cdot {0.86}^{h + x} - 90 \cdot {0.86}^{x}}{h}$

Step 4.6

Factor $90$ out of $90 \cdot {0.86}^{h + x} - 90 \cdot {0.86}^{x}$ .

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Step 4.6.1

Factor $90$ out of $90 \cdot {0.86}^{h + x}$ .

$\frac{90 \cdot {0.86}^{h + x} - 90 \cdot {0.86}^{x}}{h}$

Step 4.6.2

Factor $90$ out of $- 90 \cdot {0.86}^{x}$ .

$\frac{90 \cdot {0.86}^{h + x} + 90 (- {0.86}^{x})}{h}$

Step 4.6.3

Factor $90$ out of $90 \cdot {0.86}^{h + x} + 90 (- {0.86}^{x})$ .

$\frac{90 ({0.86}^{h + x} - {0.86}^{x})}{h}$

Step 5