Precalculus Examples

$C (x) = \frac{100}{100 - x}$

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.

$C (x + h) = \frac{100}{100 - (x + h)}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$C (x + h) = \frac{100}{100 - x - h}$

Step 2.1.2.2

The final answer is $\frac{100}{100 - x - h}$ .

$\frac{100}{100 - x - h}$

Step 2.2

Find the components of the definition.

$C (x + h) = \frac{100}{100 - x - h}$

$C (x) = \frac{100}{100 - x}$

$C (x + h) = \frac{100}{100 - x - h}$

$C (x) = \frac{100}{100 - x}$

Step 3

Plug in the components.

$\frac{C (x + h) - C x}{h} = \frac{\frac{100}{100 - x - h} - (\frac{100}{100 - x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{100}{100 - x - h}$ as a fraction with a common denominator, multiply by $\frac{100 - x}{100 - x}$ .

$\frac{\frac{100}{100 - x - h} \cdot \frac{100 - x}{100 - x} - \frac{100}{100 - x}}{h}$

Step 4.1.2

To write $- \frac{100}{100 - x}$ as a fraction with a common denominator, multiply by $\frac{100 - x - h}{100 - x - h}$ .

$\frac{\frac{100}{100 - x - h} \cdot \frac{100 - x}{100 - x} - \frac{100}{100 - x} \cdot \frac{100 - x - h}{100 - x - h}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(100 - x - h) (100 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{100}{100 - x - h}$ by $\frac{100 - x}{100 - x}$ .

$\frac{\frac{100 (100 - x)}{(100 - x - h) (100 - x)} - \frac{100}{100 - x} \cdot \frac{100 - x - h}{100 - x - h}}{h}$

Step 4.1.3.2

Multiply $\frac{100}{100 - x}$ by $\frac{100 - x - h}{100 - x - h}$ .

$\frac{\frac{100 (100 - x)}{(100 - x - h) (100 - x)} - \frac{100 (100 - x - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.3.3

Reorder the factors of $(100 - x - h) (100 - x)$ .

$\frac{\frac{100 (100 - x)}{(100 - x) (100 - x - h)} - \frac{100 (100 - x - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{100 (100 - x) - 100 (100 - x - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5

Rewrite $\frac{100 (100 - x) - 100 (100 - x - h)}{(100 - x) (100 - x - h)}$ in a factored form.

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Step 4.1.5.1

Factor $100$ out of $100 (100 - x) - 100 (100 - x - h)$ .

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Step 4.1.5.1.1

Factor $100$ out of $- 100 (100 - x - h)$ .

$\frac{\frac{100 (100 - x) + 100 (- (100 - x - h))}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.1.2

Factor $100$ out of $100 (100 - x) + 100 (- (100 - x - h))$ .

$\frac{\frac{100 (100 - x - (100 - x - h))}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.2

Apply the distributive property.

$\frac{\frac{100 (100 - x - 1 \cdot 100 - - x - - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.3

Simplify.

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Step 4.1.5.3.1

Multiply $- 1$ by $100$ .

$\frac{\frac{100 (100 - x - 100 - - x - - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.3.2

Multiply $- - x$ .

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Step 4.1.5.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{100 (100 - x - 100 + 1 x - - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.3.2.2

Multiply $x$ by $1$ .

$\frac{\frac{100 (100 - x - 100 + x - - h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.3.3

Multiply $- - h$ .

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Step 4.1.5.3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{100 (100 - x - 100 + x + 1 h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.3.3.2

Multiply $h$ by $1$ .

$\frac{\frac{100 (100 - x - 100 + x + h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.4

Subtract $100$ from $100$ .

$\frac{\frac{100 (- x + 0 + x + h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.5

Add $- x$ and $0$ .

$\frac{\frac{100 (- x + x + h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.6

Add $- x$ and $x$ .

$\frac{\frac{100 (0 + h)}{(100 - x) (100 - x - h)}}{h}$

Step 4.1.5.7

Add $0$ and $h$ .

$\frac{\frac{100 h}{(100 - x) (100 - x - h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{100 h}{(100 - x) (100 - x - h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $100 h$ .

$\frac{h \cdot 100}{(100 - x) (100 - x - h)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 100}{(100 - x) (100 - x - h)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{100}{(100 - x) (100 - x - h)}$

Step 5