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Examples

$f (x) = 3 x + 3$ , $x = 3$

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.

$f (x + h) = 3 (x + h) + 3$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 3 x + 3 h + 3$

Step 2.1.2.2

The final answer is $3 x + 3 h + 3$ .

$3 x + 3 h + 3$

$3 x + 3 h + 3$

$3 x + 3 h + 3$

Step 2.2

Reorder $3 x$ and $3 h$ .

$3 h + 3 x + 3$

Step 2.3

Find the components of the definition.

$f (x + h) = 3 h + 3 x + 3$

$f (x) = 3 x + 3$

$f (x + h) = 3 h + 3 x + 3$

$f (x) = 3 x + 3$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $3$ out of $3 x + 3$ .

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Step 4.1.1.1

Factor $3$ out of $3 x$ .

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

Step 4.1.1.2

Factor $3$ out of $3$ .

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

Step 4.1.1.3

Factor $3$ out of $3 (x) + 3 (1)$ .

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

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

Step 4.1.2

Multiply $- 1$ by $3$ .

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

Step 4.1.3

Factor $3$ out of $3 h + 3 x + 3 - 3 (x + 1)$ .

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Step 4.1.3.1

Factor $3$ out of $3 h$ .

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

Step 4.1.3.2

Factor $3$ out of $3 x$ .

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

Step 4.1.3.3

Factor $3$ out of $3$ .

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

Step 4.1.3.4

Factor $3$ out of $- 3 (x + 1)$ .

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

Step 4.1.3.5

Factor $3$ out of $3 h + 3 (x)$ .

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

Step 4.1.3.6

Factor $3$ out of $3 (h + x) + 3 (1)$ .

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

Step 4.1.3.7

Factor $3$ out of $3 (h + x + 1) + 3 (- (x + 1))$ .

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

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

Step 4.1.4

Apply the distributive property.

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

Step 4.1.5

Multiply $- 1$ by $1$ .

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

Step 4.1.6

Subtract $x$ from $x$ .

$\frac{3 (h + 0 + 1 - 1)}{h}$

Step 4.1.7

Add $h$ and $0$ .

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

Step 4.1.8

Subtract $1$ from $1$ .

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

Step 4.1.9

Add $h$ and $0$ .

$\frac{3 h}{h}$

$\frac{3 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{3 h}{h}$

Step 4.2.2

Divide $3$ by $1$ .

$3$

$3$

$3$

Step 5

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