Precalculus Examples

f (x) = 3 x + 3

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

x = 3

$x = 3$

Step 1

Consider the difference quotient formula.

\frac{f (x + h) - f (x)}{h}

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at

x = x + h

$x = x + h$ .

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Step 2.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

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

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

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

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

Step 2.1.2.2

The final answer is

3 x + 3 h + 3

$3 x + 3 h + 3$ .

3 x + 3 h + 3

$3 x + 3 h + 3$

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

$3 x$ and

3 h

$3 h$ .

3 h + 3 x + 3

$3 h + 3 x + 3$

Step 2.3

Find the components of the definition.

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

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

f (x) = 3 x + 3

$f (x) = 3 x + 3$

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

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

f (x) = 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}

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor

3

$3$ out of

3 x + 3

$3 x + 3$ .

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Step 4.1.1.1

Factor

3

$3$ out of

3 x

$3 x$ .

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

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

Step 4.1.1.2

Factor

3

$3$ out of

3

$3$ .

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

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

Step 4.1.1.3

Factor

3

$3$ out of

3 (x) + 3 (1)

$3 (x) + 3 (1)$ .

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

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

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

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

Step 4.1.2

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 4.1.3

Factor

3

$3$ out of

3 h + 3 x + 3 - 3 (x + 1)

$3 h + 3 x + 3 - 3 (x + 1)$ .

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Step 4.1.3.1

Factor

3

$3$ out of

3 h

$3 h$ .

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

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

Step 4.1.3.2

Factor

3

$3$ out of

3 x

$3 x$ .

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

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

Step 4.1.3.3

Factor

3

$3$ out of

3

$3$ .

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

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

Step 4.1.3.4

Factor

3

$3$ out of

- 3 (x + 1)

$- 3 (x + 1)$ .

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

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

Step 4.1.3.5

Factor

3

$3$ out of

3 h + 3 (x)

$3 h + 3 (x)$ .

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

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

Step 4.1.3.6

Factor

3

$3$ out of

3 (h + x) + 3 (1)

$3 (h + x) + 3 (1)$ .

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

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

Step 4.1.3.7

Factor

3

$3$ out of

3 (h + x + 1) + 3 (- (x + 1))

$3 (h + x + 1) + 3 (- (x + 1))$ .

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

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

\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}

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

Step 4.1.5

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 4.1.6

Subtract

x

$x$ from

x

$x$ .

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

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

Step 4.1.7

Add

h

$h$ and

0

$0$ .

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

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

Step 4.1.8

Subtract

1

$1$ from

1

$1$ .

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

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

Step 4.1.9

Add

h

$h$ and

0

$0$ .

\frac{3 h}{h}

$\frac{3 h}{h}$

\frac{3 h}{h}

$\frac{3 h}{h}$

Step 4.2

Cancel the common factor of

h

$h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{3 h}{h}$

Step 4.2.2

Divide

$3$ by

$1$ .

$3$

Step 5

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