Precalculus Examples

$f (x) = 2 {(\frac{5}{2})}^{- x}$

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.

$f (x + h) = 2 {(\frac{5}{2})}^{- (x + h)}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 2 {(\frac{5}{2})}^{- x - h}$

Step 2.1.2.2

Apply the product rule to $\frac{5}{2}$ .

$f (x + h) = 2 (\frac{5^{- x - h}}{2^{- x - h}})$

Step 2.1.2.3

Combine $2$ and $\frac{5^{- x - h}}{2^{- x - h}}$ .

$f (x + h) = \frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$

Step 2.1.2.4

The final answer is $\frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$ .

$\frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$

$f (x) = \frac{2 \cdot 5^{- x}}{2^{- x}}$

$f (x + h) = \frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$

$f (x) = \frac{2 \cdot 5^{- x}}{2^{- x}}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{2 \cdot 5^{- x - h}}{2^{- x - h}} - (\frac{2 \cdot 5^{- x}}{2^{- x}})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

To write $\frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$ as a fraction with a common denominator, multiply by $\frac{2^{- x}}{2^{- x}}$ .

$\frac{\frac{2 \cdot 5^{- x - h}}{2^{- x - h}} \cdot \frac{2^{- x}}{2^{- x}} - \frac{2 \cdot 5^{- x}}{2^{- x}}}{h}$

Step 4.2

To write $- \frac{2 \cdot 5^{- x}}{2^{- x}}$ as a fraction with a common denominator, multiply by $\frac{2^{- h - x}}{2^{- h - x}}$ .

$\frac{\frac{2 \cdot 5^{- x - h}}{2^{- x - h}} \cdot \frac{2^{- x}}{2^{- x}} - \frac{2 \cdot 5^{- x}}{2^{- x}} \cdot \frac{2^{- h - x}}{2^{- h - x}}}{h}$

Step 4.3

Write each expression with a common denominator of $2^{- x - h} \cdot 2^{- x}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{2 \cdot 5^{- x - h}}{2^{- x - h}}$ by $\frac{2^{- x}}{2^{- x}}$ .

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- x - h} \cdot 2^{- x}} - \frac{2 \cdot 5^{- x}}{2^{- x}} \cdot \frac{2^{- h - x}}{2^{- h - x}}}{h}$

Step 4.3.2

Multiply $2^{- x - h}$ by $2^{- x}$ by adding the exponents.

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Step 4.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- x - h - x}} - \frac{2 \cdot 5^{- x}}{2^{- x}} \cdot \frac{2^{- h - x}}{2^{- h - x}}}{h}$

Step 4.3.2.2

Subtract $x$ from $- x$ .

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- 2 x - h}} - \frac{2 \cdot 5^{- x}}{2^{- x}} \cdot \frac{2^{- h - x}}{2^{- h - x}}}{h}$

Step 4.3.3

Multiply $\frac{2 \cdot 5^{- x}}{2^{- x}}$ by $\frac{2^{- h - x}}{2^{- h - x}}$ .

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- 2 x - h}} - \frac{2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- x} \cdot 2^{- h - x}}}{h}$

Step 4.3.4

Multiply $2^{- x}$ by $2^{- h - x}$ by adding the exponents.

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Step 4.3.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- 2 x - h}} - \frac{2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- x - h - x}}}{h}$

Step 4.3.4.2

Subtract $x$ from $- x$ .

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x}}{2^{- 2 x - h}} - \frac{2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- 2 x - h}}}{h}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 5^{- x - h} \cdot 2^{- x} - 2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- 2 x - h}}}{h}$

Step 4.5

Rewrite $\frac{2 \cdot 5^{- x - h} \cdot 2^{- x} - 2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- 2 x - h}}$ in a factored form.

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Step 4.5.1

Factor $2$ out of $2 \cdot 5^{- x - h} \cdot 2^{- x} - 2 \cdot 5^{- x} \cdot 2^{- h - x}$ .

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Step 4.5.1.1

Factor $2$ out of $2 \cdot 5^{- x - h} \cdot 2^{- x}$ .

$\frac{\frac{2 (5^{- x - h} \cdot 2^{- x}) - 2 \cdot 5^{- x} \cdot 2^{- h - x}}{2^{- 2 x - h}}}{h}$

Step 4.5.1.2

Factor $2$ out of $- 2 \cdot 5^{- x} \cdot 2^{- h - x}$ .

$\frac{\frac{2 (5^{- x - h} \cdot 2^{- x}) + 2 (- 5^{- x} \cdot 2^{- h - x})}{2^{- 2 x - h}}}{h}$

Step 4.5.1.3

Factor $2$ out of $2 (5^{- x - h} \cdot 2^{- x}) + 2 (- 5^{- x} \cdot 2^{- h - x})$ .

$\frac{\frac{2 (5^{- x - h} \cdot 2^{- x} - 5^{- x} \cdot 2^{- h - x})}{2^{- 2 x - h}}}{h}$

Step 4.5.2

Reduce the expression by cancelling the common factors.

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Step 4.5.2.1

Reduce the expression $\frac{2 (5^{- x - h} \cdot 2^{- x} - 5^{- x} \cdot 2^{- h - x})}{2^{- 2 x - h}}$ by cancelling the common factors.

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Step 4.5.2.1.1

Factor $2^{- 2 x - h}$ out of $2 (5^{- x - h} \cdot 2^{- x} - 5^{- x} \cdot 2^{- h - x})$ .

$\frac{\frac{2^{- 2 x - h} (2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)}))}{2^{- 2 x - h}}}{h}$

Step 4.5.2.1.2

Multiply by $1$ .

$\frac{\frac{2^{- 2 x - h} (2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)}))}{2^{- 2 x - h} \cdot 1}}{h}$

Step 4.5.2.1.3

Cancel the common factor.

$\frac{\frac{2^{- 2 x - h} (2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)}))}{2^{- 2 x - h} \cdot 1}}{h}$

Step 4.5.2.1.4

Rewrite the expression.

$\frac{\frac{2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{1}}{h}$

Step 4.5.2.2

Divide $2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})$ by $1$ .

$\frac{2 (5^{- x - h} \cdot 2^{- x - (- 2 x - h)} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6

Simplify each term.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

Apply the distributive property.

$\frac{2 (5^{- x - h} \cdot 2^{- x - (- 2 x) - - h} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6.1.2

Multiply $- 2$ by $- 1$ .

$\frac{2 (5^{- x - h} \cdot 2^{- x + 2 x - - h} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6.1.3

Multiply $- - h$ .

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Step 4.6.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (5^{- x - h} \cdot 2^{- x + 2 x + 1 h} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6.1.3.2

Multiply $h$ by $1$ .

$\frac{2 (5^{- x - h} \cdot 2^{- x + 2 x + h} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6.2

Add $- x$ and $2 x$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- h - x - (- 2 x - h)})}{h}$

Step 4.6.3

Simplify each term.

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Step 4.6.3.1

Apply the distributive property.

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- h - x - (- 2 x) - - h})}{h}$

Step 4.6.3.2

Multiply $- 2$ by $- 1$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- h - x + 2 x - - h})}{h}$

Step 4.6.3.3

Multiply $- - h$ .

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Step 4.6.3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- h - x + 2 x + 1 h})}{h}$

Step 4.6.3.3.2

Multiply $h$ by $1$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- h - x + 2 x + h})}{h}$

Step 4.6.4

Combine the opposite terms in $- h - x + 2 x + h$ .

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Step 4.6.4.1

Add $- h$ and $h$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- x + 2 x + 0})}{h}$

Step 4.6.4.2

Add $- x + 2 x$ and $0$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{- x + 2 x})}{h}$

Step 4.6.5

Add $- x$ and $2 x$ .

$\frac{2 (5^{- x - h} \cdot 2^{x + h} - 5^{- x} \cdot 2^{x})}{h}$

Step 5