Precalculus Examples

$f (x) = - 2 e^{x + 3} + 2$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

The final answer is $- 2 e^{x + h + 3} + 2$ .

$- 2 e^{x + h + 3} + 2$

Step 2.2

Find the components of the definition.

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

$f (x) = - 2 e^{x + 3} + 2$

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

$f (x) = - 2 e^{x + 3} + 2$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $- 2$ by $- 1$ .

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

Step 4.1.3

Multiply $- 1$ by $2$ .

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

Step 4.1.4

Subtract $2$ from $2$ .

$\frac{- 2 e^{x + h + 3} + 2 e^{x + 3} + 0}{h}$

Step 4.1.5

Add $- 2 e^{x + h + 3} + 2 e^{x + 3}$ and $0$ .

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

Step 4.1.6

Factor $2$ out of $- 2 e^{x + h + 3} + 2 e^{x + 3}$ .

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Step 4.1.6.1

Factor $2$ out of $- 2 e^{x + h + 3}$ .

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

Step 4.1.6.2

Factor $2$ out of $2 e^{x + 3}$ .

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

Step 4.1.6.3

Factor $2$ out of $2 (- e^{x + h + 3}) + 2 (e^{x + 3})$ .

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

Step 4.2

Simplify with factoring out.

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Step 4.2.1

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

$\frac{2 (- e^{x + h + 3} - 1 (- e^{x + 3}))}{h}$

Step 4.2.2

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

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

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Rewrite $- (e^{x + h + 3} - e^{x + 3})$ as $- 1 (e^{x + h + 3} - e^{x + 3})$ .

$\frac{2 (- 1 (e^{x + h + 3} - e^{x + 3}))}{h}$

Step 4.2.3.2

Move the negative in front of the fraction.

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

Step 5