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Examples

$f (x) = 4 x - 4$ , $x = 2$

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.

$f (x + h) = 4 (x + h) - 4$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 4 x + 4 h - 4$

Step 2.1.2.2

The final answer is $4 x + 4 h - 4$ .

$4 x + 4 h - 4$

$4 x + 4 h - 4$

$4 x + 4 h - 4$

Step 2.2

Reorder $4 x$ and $4 h$ .

$4 h + 4 x - 4$

Step 2.3

Find the components of the definition.

$f (x + h) = 4 h + 4 x - 4$

$f (x) = 4 x - 4$

$f (x + h) = 4 h + 4 x - 4$

$f (x) = 4 x - 4$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $4$ out of $4 x - 4$ .

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Step 4.1.1.1

Factor $4$ out of $4 x$ .

$\frac{4 h + 4 x - 4 - (4 (x) - 4)}{h}$

Step 4.1.1.2

Factor $4$ out of $- 4$ .

$\frac{4 h + 4 x - 4 - (4 (x) + 4 (- 1))}{h}$

Step 4.1.1.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

$\frac{4 h + 4 x - 4 - (4 (x - 1))}{h}$

$\frac{4 h + 4 x - 4 - 1 \cdot 4 (x - 1)}{h}$

Step 4.1.2

Multiply $- 1$ by $4$ .

$\frac{4 h + 4 x - 4 - 4 (x - 1)}{h}$

Step 4.1.3

Factor $4$ out of $4 h + 4 x - 4 - 4 (x - 1)$ .

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Step 4.1.3.1

Factor $4$ out of $4 h$ .

$\frac{4 h + 4 x - 4 - 4 (x - 1)}{h}$

Step 4.1.3.2

Factor $4$ out of $4 x$ .

$\frac{4 h + 4 (x) - 4 - 4 (x - 1)}{h}$

Step 4.1.3.3

Factor $4$ out of $- 4$ .

$\frac{4 h + 4 (x) + 4 (- 1) - 4 (x - 1)}{h}$

Step 4.1.3.4

Factor $4$ out of $- 4 (x - 1)$ .

$\frac{4 h + 4 (x) + 4 (- 1) + 4 (- (x - 1))}{h}$

Step 4.1.3.5

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

$\frac{4 (h + x) + 4 (- 1) + 4 (- (x - 1))}{h}$

Step 4.1.3.6

Factor $4$ out of $4 (h + x) + 4 (- 1)$ .

$\frac{4 (h + x - 1) + 4 (- (x - 1))}{h}$

Step 4.1.3.7

Factor $4$ out of $4 (h + x - 1) + 4 (- (x - 1))$ .

$\frac{4 (h + x - 1 - (x - 1))}{h}$

$\frac{4 (h + x - 1 - (x - 1))}{h}$

Step 4.1.4

Apply the distributive property.

$\frac{4 (h + x - 1 - x - - 1)}{h}$

Step 4.1.5

Multiply $- 1$ by $- 1$ .

$\frac{4 (h + x - 1 - x + 1)}{h}$

Step 4.1.6

Subtract $x$ from $x$ .

$\frac{4 (h + 0 - 1 + 1)}{h}$

Step 4.1.7

Add $h$ and $0$ .

$\frac{4 (h - 1 + 1)}{h}$

Step 4.1.8

Add $- 1$ and $1$ .

$\frac{4 (h + 0)}{h}$

Step 4.1.9

Add $h$ and $0$ .

$\frac{4 h}{h}$

$\frac{4 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{4 h}{h}$

Step 4.2.2

Divide $4$ by $1$ .

$4$

$4$

$4$

Step 5

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$