Precalculus Examples

$f (x) = \frac{4 x - 2}{3 x}$

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.

$f (x + h) = \frac{4 (x + h) - 2}{3 (x + h)}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Factor $2$ out of $4 (x + h) - 2$ .

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Step 2.1.2.1.1.1

Factor $2$ out of $4 (x + h)$ .

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

Step 2.1.2.1.1.2

Factor $2$ out of $- 2$ .

$f (x + h) = \frac{2 (2 (x + h)) + 2 (- 1)}{3 (x + h)}$

Step 2.1.2.1.1.3

Factor $2$ out of $2 (2 (x + h)) + 2 (- 1)$ .

$f (x + h) = \frac{2 (2 (x + h) - 1)}{3 (x + h)}$

Step 2.1.2.1.2

Apply the distributive property.

$f (x + h) = \frac{2 (2 x + 2 h - 1)}{3 (x + h)}$

Step 2.1.2.2

The final answer is $\frac{2 (2 x + 2 h - 1)}{3 (x + h)}$ .

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

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{2 (2 x + 2 h - 1)}{3 (x + h)}$

$f (x) = \frac{2 (2 x - 1)}{3 x}$

$f (x + h) = \frac{2 (2 x + 2 h - 1)}{3 (x + h)}$

$f (x) = \frac{2 (2 x - 1)}{3 x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{2 (2 x + 2 h - 1)}{3 (x + h)} - (\frac{2 (2 x - 1)}{3 x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{2 (2 x + 2 h - 1)}{3 (x + h)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.1.2

To write $- \frac{2 (2 x - 1)}{3 x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

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

Step 4.1.3

Write each expression with a common denominator of $3 (x + h) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{2 (2 x + 2 h - 1)}{3 (x + h)}$ by $\frac{x}{x}$ .

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

Step 4.1.3.2

Multiply $\frac{2 (2 x - 1)}{3 x}$ by $\frac{x + h}{x + h}$ .

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

Step 4.1.3.3

Reorder the factors of $3 (x + h) x$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{2 (2 x + 2 h - 1) x - 2 (2 x - 1) (x + h)}{3 x (x + h)}$ in a factored form.

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Step 4.1.5.1

Factor $2$ out of $2 (2 x + 2 h - 1) x - 2 (2 x - 1) (x + h)$ .

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Step 4.1.5.1.1

Factor $2$ out of $2 (2 x + 2 h - 1) x$ .

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

Step 4.1.5.1.2

Factor $2$ out of $- 2 (2 x - 1) (x + h)$ .

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

Step 4.1.5.1.3

Factor $2$ out of $2 ((2 x + 2 h - 1) x) + 2 (- (2 x - 1) (x + h))$ .

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

Step 4.1.5.2

Apply the distributive property.

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

Step 4.1.5.3

Simplify.

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Step 4.1.5.3.1

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

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Step 4.1.5.3.1.1

Move $x$ .

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

Step 4.1.5.3.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.3.2

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

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

Step 4.1.5.4

Apply the distributive property.

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

Step 4.1.5.5

Multiply $2$ by $- 1$ .

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

Step 4.1.5.6

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.7

Expand $(- 2 x + 1) (x + h)$ using the FOIL Method.

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Step 4.1.5.7.1

Apply the distributive property.

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

Step 4.1.5.7.2

Apply the distributive property.

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

Step 4.1.5.7.3

Apply the distributive property.

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

Step 4.1.5.8

Simplify each term.

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Step 4.1.5.8.1

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

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Step 4.1.5.8.1.1

Move $x$ .

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

Step 4.1.5.8.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.8.2

Multiply $x$ by $1$ .

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

Step 4.1.5.8.3

Multiply $h$ by $1$ .

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

Step 4.1.5.9

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

$\frac{\frac{2 (2 h x - x + 0 - 2 x h + x + h)}{3 x (x + h)}}{h}$

Step 4.1.5.10

Add $2 h x$ and $0$ .

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

Step 4.1.5.11

Subtract $2 x h$ from $2 h x$ .

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Step 4.1.5.11.1

Move $h$ .

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

Step 4.1.5.11.2

Subtract $2 x h$ from $2 x h$ .

$\frac{\frac{2 (- x + 0 + x + h)}{3 x (x + h)}}{h}$

Step 4.1.5.12

Add $- x$ and $0$ .

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

Step 4.1.5.13

Add $- x$ and $x$ .

$\frac{\frac{2 (0 + h)}{3 x (x + h)}}{h}$

Step 4.1.5.14

Add $0$ and $h$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $2 h$ .

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 5