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Precalculus Examples

$f (x) = 6 x + 6$

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.

$f (x + h) = 6 (x + h) + 6$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 6 x + 6 h + 6$

Step 2.1.2.2

The final answer is $6 x + 6 h + 6$ .

$6 x + 6 h + 6$

$6 x + 6 h + 6$

$6 x + 6 h + 6$

Step 2.2

Reorder $6 x$ and $6 h$ .

$6 h + 6 x + 6$

Step 2.3

Find the components of the definition.

$f (x + h) = 6 h + 6 x + 6$

$f (x) = 6 x + 6$

$f (x + h) = 6 h + 6 x + 6$

$f (x) = 6 x + 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{6 h + 6 x + 6 - (6 x + 6)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $6$ out of $6 x + 6$ .

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Step 4.1.1.1

Factor $6$ out of $6 x$ .

$\frac{6 h + 6 x + 6 - (6 (x) + 6)}{h}$

Step 4.1.1.2

Factor $6$ out of $6$ .

$\frac{6 h + 6 x + 6 - (6 (x) + 6 (1))}{h}$

Step 4.1.1.3

Factor $6$ out of $6 (x) + 6 (1)$ .

$\frac{6 h + 6 x + 6 - (6 (x + 1))}{h}$

$\frac{6 h + 6 x + 6 - 1 \cdot 6 (x + 1)}{h}$

Step 4.1.2

Multiply $- 1$ by $6$ .

$\frac{6 h + 6 x + 6 - 6 (x + 1)}{h}$

Step 4.1.3

Factor $6$ out of $6 h + 6 x + 6 - 6 (x + 1)$ .

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Step 4.1.3.1

Factor $6$ out of $6 h$ .

$\frac{6 h + 6 x + 6 - 6 (x + 1)}{h}$

Step 4.1.3.2

Factor $6$ out of $6 x$ .

$\frac{6 h + 6 (x) + 6 - 6 (x + 1)}{h}$

Step 4.1.3.3

Factor $6$ out of $6$ .

$\frac{6 h + 6 (x) + 6 (1) - 6 (x + 1)}{h}$

Step 4.1.3.4

Factor $6$ out of $- 6 (x + 1)$ .

$\frac{6 h + 6 (x) + 6 (1) + 6 (- (x + 1))}{h}$

Step 4.1.3.5

Factor $6$ out of $6 h + 6 (x)$ .

$\frac{6 (h + x) + 6 (1) + 6 (- (x + 1))}{h}$

Step 4.1.3.6

Factor $6$ out of $6 (h + x) + 6 (1)$ .

$\frac{6 (h + x + 1) + 6 (- (x + 1))}{h}$

Step 4.1.3.7

Factor $6$ out of $6 (h + x + 1) + 6 (- (x + 1))$ .

$\frac{6 (h + x + 1 - (x + 1))}{h}$

$\frac{6 (h + x + 1 - (x + 1))}{h}$

Step 4.1.4

Apply the distributive property.

$\frac{6 (h + x + 1 - x - 1 \cdot 1)}{h}$

Step 4.1.5

Multiply $- 1$ by $1$ .

$\frac{6 (h + x + 1 - x - 1)}{h}$

Step 4.1.6

Subtract $x$ from $x$ .

$\frac{6 (h + 0 + 1 - 1)}{h}$

Step 4.1.7

Add $h$ and $0$ .

$\frac{6 (h + 1 - 1)}{h}$

Step 4.1.8

Subtract $1$ from $1$ .

$\frac{6 (h + 0)}{h}$

Step 4.1.9

Add $h$ and $0$ .

$\frac{6 h}{h}$

$\frac{6 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{6 h}{h}$

Step 4.2.2

Divide $6$ by $1$ .

$6$

$6$

$6$

Step 5

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