Precalculus Examples

$f (x) = 3 x^{2} + 6$

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.

$f (x + h) = 3 {(x + h)}^{2} + 6$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite

${(x + h)}^{2}$ as

$(x + h) (x + h)$ .

$f (x + h) = 3 ((x + h) (x + h)) + 6$

Step 2.1.2.1.2

Expand

$(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 3 (x (x + h) + h (x + h)) + 6$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h (x + h)) + 6$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 3 (x \cdot x + x h + h x + h \cdot h) + 6$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply

$x$ by

$x$ .

$f (x + h) = 3 (x^{2} + x h + h x + h \cdot h) + 6$

Step 2.1.2.1.3.1.2

Multiply

$h$ by

$h$ .

$f (x + h) = 3 (x^{2} + x h + h x + h^{2}) + 6$

Step 2.1.2.1.3.2

Add

$x h$ and

$h x$ .

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Step 2.1.2.1.3.2.1

Reorder

$x$ and

$h$ .

$f (x + h) = 3 (x^{2} + h x + h x + h^{2}) + 6$

Step 2.1.2.1.3.2.2

Add

$h x$ and

$h x$ .

$f (x + h) = 3 (x^{2} + 2 h x + h^{2}) + 6$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 3 x^{2} + 3 (2 h x) + 3 h^{2} + 6$

Step 2.1.2.1.5

Multiply

$2$ by

$3$ .

$f (x + h) = 3 x^{2} + 6 h x + 3 h^{2} + 6$

Step 2.1.2.2

The final answer is

$3 x^{2} + 6 h x + 3 h^{2} + 6$ .

$3 x^{2} + 6 h x + 3 h^{2} + 6$

Step 2.2

Reorder.

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Step 2.2.1

Move

$3 x^{2}$ .

$6 h x + 3 h^{2} + 3 x^{2} + 6$

Step 2.2.2

Reorder

$6 h x$ and

$3 h^{2}$ .

$3 h^{2} + 6 h x + 3 x^{2} + 6$

Step 2.3

Find the components of the definition.

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} + 6$

$f (x) = 3 x^{2} + 6$

$f (x + h) = 3 h^{2} + 6 h x + 3 x^{2} + 6$

$f (x) = 3 x^{2} + 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{3 h^{2} + 6 h x + 3 x^{2} + 6 - (3 x^{2} + 6)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{3 h^{2} + 6 h x + 3 x^{2} + 6 - (3 x^{2}) - 1 \cdot 6}{h}$

Step 4.1.2

Multiply

$3$ by

$- 1$ .

$\frac{3 h^{2} + 6 h x + 3 x^{2} + 6 - 3 x^{2} - 1 \cdot 6}{h}$

Step 4.1.3

Multiply

$- 1$ by

$6$ .

$\frac{3 h^{2} + 6 h x + 3 x^{2} + 6 - 3 x^{2} - 6}{h}$

Step 4.1.4

Subtract

$3 x^{2}$ from

$3 x^{2}$ .

$\frac{3 h^{2} + 6 h x + 0 + 6 - 6}{h}$

Step 4.1.5

Add

$3 h^{2}$ and

$0$ .

$\frac{3 h^{2} + 6 h x + 6 - 6}{h}$

Step 4.1.6

Subtract

$6$ from

$6$ .

$\frac{3 h^{2} + 6 h x + 0}{h}$

Step 4.1.7

Add

$3 h^{2} + 6 h x$ and

$0$ .

$\frac{3 h^{2} + 6 h x}{h}$

Step 4.1.8

Factor

$3 h$ out of

$3 h^{2} + 6 h x$ .

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Step 4.1.8.1

Factor

$3 h$ out of

$3 h^{2}$ .

$\frac{3 h \cdot h + 6 h x}{h}$

Step 4.1.8.2

Factor

$3 h$ out of

$6 h x$ .

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

Step 4.1.8.3

Factor

$3 h$ out of

$3 h \cdot h + 3 h (2 x)$ .

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

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of

$h$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide

$3 (h + 2 x)$ by

$1$ .

$3 (h + 2 x)$

Step 4.2.2

Apply the distributive property.

$3 h + 3 (2 x)$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Multiply

$2$ by

$3$ .

$3 h + 6 x$

Step 4.2.3.2

Reorder

$3 h$ and

$6 x$ .

$6 x + 3 h$

Step 5

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