Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Expand $(x + h + 3) (x + h - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2.1.2.2

Simplify terms.

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Step 2.1.2.2.1

Combine the opposite terms in $x \cdot x + x h + x \cdot - 3 + h x + h \cdot h + h \cdot - 3 + 3 x + 3 h + 3 \cdot - 3$ .

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Step 2.1.2.2.1.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 2.1.2.2.1.2

Add $- 3 x$ and $3 x$ .

$f (x + h) = x \cdot x + x h + h x + h \cdot h + h \cdot - 3 + 0 + 3 h + 3 \cdot - 3$

Step 2.1.2.2.1.3

Add $x \cdot x + x h + h x + h \cdot h + h \cdot - 3$ and $0$ .

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

Step 2.1.2.2.1.4

Reorder the factors in the terms $h \cdot - 3$ and $3 h$ .

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

Step 2.1.2.2.1.5

Add $- 3 h$ and $3 h$ .

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 0 + 3 \cdot - 3$

Step 2.1.2.2.1.6

Add $x \cdot x + x h + h x + h \cdot h$ and $0$ .

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

Step 2.1.2.2.2

Simplify each term.

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Step 2.1.2.2.2.1

Multiply $x$ by $x$ .

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

Step 2.1.2.2.2.2

Multiply $h$ by $h$ .

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

Step 2.1.2.2.2.3

Multiply $3$ by $- 3$ .

$f (x + h) = x^{2} + x h + h x + h^{2} - 9$

Step 2.1.2.3

Add $x h$ and $h x$ .

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Step 2.1.2.3.1

Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} - 9$

Step 2.1.2.3.2

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} - 9$

Step 2.1.2.4

The final answer is $x^{2} + 2 h x + h^{2} - 9$ .

$x^{2} + 2 h x + h^{2} - 9$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} - 9$

Step 2.2.2

Reorder $2 h x$ and $h^{2}$ .

$h^{2} + 2 h x + x^{2} - 9$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} - 9$

$f (x) = x^{2} - 9$

$f (x + h) = h^{2} + 2 h x + x^{2} - 9$

$f (x) = x^{2} - 9$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{2} + 2 h x + x^{2} - 9 - (x^{2} - 9)}{h}$

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.

$\frac{h^{2} + 2 h x + x^{2} - 9 - x^{2} - - 9}{h}$

Step 4.1.2

Multiply $- 1$ by $- 9$ .

$\frac{h^{2} + 2 h x + x^{2} - 9 - x^{2} + 9}{h}$

Step 4.1.3

Subtract $x^{2}$ from $x^{2}$ .

$\frac{h^{2} + 2 h x + 0 - 9 + 9}{h}$

Step 4.1.4

Add $h^{2}$ and $0$ .

$\frac{h^{2} + 2 h x - 9 + 9}{h}$

Step 4.1.5

Add $- 9$ and $9$ .

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

Step 4.1.6

Add $h^{2} + 2 h x$ and $0$ .

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

Step 4.1.7

Factor $h$ out of $h^{2} + 2 h x$ .

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Step 4.1.7.1

Factor $h$ out of $h^{2}$ .

$\frac{h \cdot h + 2 h x}{h}$

Step 4.1.7.2

Factor $h$ out of $2 h x$ .

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

Step 4.1.7.3

Factor $h$ out of $h (h) + h (2 x)$ .

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

Step 4.2

Reduce the expression by cancelling the common factors.

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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.

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

Step 4.2.1.2

Divide $h + 2 x$ by $1$ .

$h + 2 x$

Step 4.2.2

Reorder $h$ and $2 x$ .

$2 x + h$

Step 5