Precalculus Examples

$f (x) = 3 e^{x - 1} + 3$

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.

$f (x + h) = 3 e^{(x + h) - 1} + 3$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = 3 e^{x + h - 1} + 3$

Step 2.1.2.2

The final answer is $3 e^{x + h - 1} + 3$ .

$3 e^{x + h - 1} + 3$

Step 2.2

Find the components of the definition.

$f (x + h) = 3 e^{x + h - 1} + 3$

$f (x) = 3 e^{x - 1} + 3$

$f (x + h) = 3 e^{x + h - 1} + 3$

$f (x) = 3 e^{x - 1} + 3$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{3 e^{x + h - 1} + 3 - (3 e^{x - 1} + 3)}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 e^{x + h - 1} + 3 - (3 e^{x - 1}) - 1 \cdot 3}{h}$

Step 4.2

Multiply $3$ by $- 1$ .

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

Step 4.3

Multiply $- 1$ by $3$ .

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

Step 4.4

Subtract $3$ from $3$ .

$\frac{3 e^{x + h - 1} + 0 - 3 e^{x - 1}}{h}$

Step 4.5

Add $3 e^{x + h - 1}$ and $0$ .

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

Step 4.6

Factor $3$ out of $3 e^{x + h - 1} - 3 e^{x - 1}$ .

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Step 4.6.1

Factor $3$ out of $3 e^{x + h - 1}$ .

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

Step 4.6.2

Factor $3$ out of $- 3 e^{x - 1}$ .

$\frac{3 e^{x + h - 1} + 3 (- e^{x - 1})}{h}$

Step 4.6.3

Factor $3$ out of $3 e^{x + h - 1} + 3 (- e^{x - 1})$ .

$\frac{3 (e^{x + h - 1} - e^{x - 1})}{h}$

Step 5