Precalculus Examples

$f (x) = x^{2} - 6$

Consider the difference quotient formula.

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

Find the components of the definition.

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Evaluate the function at $x = x + h$ .

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Replace the variable $x$ with $x + h$ in the expression.

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

Simplify the result.

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Simplify each term.

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

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

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

Multiply $h$ by $h$ .

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

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

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

Add $h x$ and $h x$ .

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

The final answer is $x^{2} + 2 h x + h^{2} - 6$ .

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

Find the components of the definition.

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

$f (x) = x^{2} - 6$

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

$f (x) = x^{2} - 6$

Plug in the components.

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

Simplify.

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Simplify the numerator.

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Apply the distributive property.

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

Multiply $- 1$ by $- 6$ .

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

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

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

Add $2 h x$ and $0$ .

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

Add $- 6$ and $6$ .

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

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

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

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

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

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

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

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

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

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

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

$2 x + h$

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