Precalculus Examples

$f (x) = \frac{K x}{(A + x)}$

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.

$f (x + h) = \frac{K (x + h)}{A + x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{K (x + h)}{A + x + h}$

Step 2.1.2.2

The final answer is $\frac{K (x + h)}{A + x + h}$ .

$\frac{K (x + h)}{A + x + h}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{K (x + h)}{A + x + h}$

$f (x) = \frac{K x}{A + x}$

$f (x + h) = \frac{K (x + h)}{A + x + h}$

$f (x) = \frac{K x}{A + x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{K (x + h)}{A + x + h} - (\frac{K x}{A + x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{K (x + h)}{A + x + h}$ as a fraction with a common denominator, multiply by $\frac{A + x}{A + x}$ .

$\frac{\frac{K (x + h)}{A + x + h} \cdot \frac{A + x}{A + x} - \frac{K x}{A + x}}{h}$

Step 4.1.2

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

$\frac{\frac{K (x + h)}{A + x + h} \cdot \frac{A + x}{A + x} - \frac{K x}{A + x} \cdot \frac{A + x + h}{A + x + h}}{h}$

Step 4.1.3

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

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Step 4.1.3.1

Multiply $\frac{K (x + h)}{A + x + h}$ by $\frac{A + x}{A + x}$ .

$\frac{\frac{K (x + h) (A + x)}{(A + x + h) (A + x)} - \frac{K x}{A + x} \cdot \frac{A + x + h}{A + x + h}}{h}$

Step 4.1.3.2

Multiply $\frac{K x}{A + x}$ by $\frac{A + x + h}{A + x + h}$ .

$\frac{\frac{K (x + h) (A + x)}{(A + x + h) (A + x)} - \frac{K x (A + x + h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.3.3

Reorder the factors of $(A + x + h) (A + x)$ .

$\frac{\frac{K (x + h) (A + x)}{(A + x) (A + x + h)} - \frac{K x (A + x + h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{K (x + h) (A + x) - K x (A + x + h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5

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

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Step 4.1.5.1

Factor $K$ out of $K (x + h) (A + x) - K x (A + x + h)$ .

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Step 4.1.5.1.1

Factor $K$ out of $K (x + h) (A + x)$ .

$\frac{\frac{K ((x + h) (A + x)) - K x (A + x + h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.1.2

Factor $K$ out of $- K x (A + x + h)$ .

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

Step 4.1.5.1.3

Factor $K$ out of $K ((x + h) (A + x)) + K (- 1 x (A + x + h))$ .

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

Step 4.1.5.2

Expand $(x + h) (A + x)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

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

Step 4.1.5.2.2

Apply the distributive property.

$\frac{\frac{K (x A + x \cdot x + h (A + x) - 1 x (A + x + h))}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$\frac{\frac{K (x A + x \cdot x + h A + h x - 1 x (A + x + h))}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.3

Multiply $x$ by $x$ .

$\frac{\frac{K (x A + x^{2} + h A + h x - 1 x (A + x + h))}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.4

Rewrite $- 1 x$ as $- x$ .

$\frac{\frac{K (x A + x^{2} + h A + h x - x (A + x + h))}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.5

Apply the distributive property.

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

Step 4.1.5.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.6.1

Move $x$ .

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

Step 4.1.5.6.2

Multiply $x$ by $x$ .

$\frac{\frac{K (x A + x^{2} + h A + h x - x A - x^{2} - x h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.7

Subtract $x A$ from $x A$ .

$\frac{\frac{K (x^{2} + h A + h x + 0 - x^{2} - x h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.8

Add $x^{2}$ and $0$ .

$\frac{\frac{K (x^{2} + h A + h x - x^{2} - x h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.9

Subtract $x^{2}$ from $x^{2}$ .

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

Step 4.1.5.10

Add $h A$ and $0$ .

$\frac{\frac{K (h A + h x - x h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.11

Subtract $x h$ from $h x$ .

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Step 4.1.5.11.1

Reorder $h$ and $x$ .

$\frac{\frac{K (h A + x h - x h)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.11.2

Subtract $x h$ from $x h$ .

$\frac{\frac{K (h A + 0)}{(A + x) (A + x + h)}}{h}$

Step 4.1.5.12

Add $h A$ and $0$ .

$\frac{\frac{K h A}{(A + x) (A + x + h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{K h A}{(A + x) (A + x + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $K h A$ .

$\frac{h (K A)}{(A + x) (A + x + h)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h (K A)}{(A + x) (A + x + h)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{K A}{(A + x) (A + x + h)}$

Step 5