Precalculus Examples

$f (x) = e^{2 x} - e^{x} + 1$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

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

$e^{2 x + 2 h} - e^{x + h} + 1$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $2 x$ and $2 h$ .

$e^{2 h + 2 x} - e^{x + h} + 1$

Step 2.2.2

Reorder $x$ and $h$ .

$e^{2 h + 2 x} - e^{h + x} + 1$

Step 2.3

Find the components of the definition.

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

$f (x) = e^{2 x} - e^{x} + 1$

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

$f (x) = e^{2 x} - e^{x} + 1$

Step 3

Plug in the components.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $- - e^{x}$ .

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Step 4.2.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.1.2

Multiply $e^{x}$ by $1$ .

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

Step 4.2.2

Multiply $- 1$ by $1$ .

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

Step 4.3

Subtract $1$ from $1$ .

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

Step 4.4

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

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

Step 5