Precalculus Examples

$f (x) = x^{2} + 5$

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

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

$f (x + h) = {(x + h)}^{2} + 5$

Simplify the result.

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Simplify each term.

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

$f (x + h) = (x + h) (x + h) + 5$

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

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Apply the distributive property.

$f (x + h) = x (x + h) + h (x + h) + 5$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h (x + h) + 5$

Apply the distributive property.

$f (x + h) = x \cdot x + x h + h x + h \cdot h + 5$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$f (x + h) = x^{2} + x h + h x + h \cdot h + 5$

Multiply $h$ by $h$ .

$f (x + h) = x^{2} + x h + h x + h^{2} + 5$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = x^{2} + h x + h x + h^{2} + 5$

Add $h x$ and $h x$ .

$f (x + h) = x^{2} + 2 h x + h^{2} + 5$

The final answer is $x^{2} + 2 h x + h^{2} + 5$ .

$x^{2} + 2 h x + h^{2} + 5$

Reorder.

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Move $x^{2}$ .

$2 h x + h^{2} + x^{2} + 5$

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

$h^{2} + 2 h x + x^{2} + 5$

Find the components of the definition.

$f (x + h) = h^{2} + 2 h x + x^{2} + 5$

$f (x) = x^{2} + 5$

$f (x + h) = h^{2} + 2 h x + x^{2} + 5$

$f (x) = x^{2} + 5$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{2} + 2 h x + x^{2} + 5 - (x^{2} + 5)}{h}$

Step 4

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$\frac{h^{2} + 2 h x + x^{2} + 5 - x^{2} - 1 \cdot 5}{h}$

Multiply $- 1$ by $5$ .

$\frac{h^{2} + 2 h x + x^{2} + 5 - x^{2} - 5}{h}$

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

$\frac{h^{2} + 2 h x + 0 + 5 - 5}{h}$

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

$\frac{h^{2} + 2 h x + 5 - 5}{h}$

Subtract $5$ from $5$ .

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

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

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

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

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

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

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

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

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

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

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $h$ .

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Cancel the common factor.

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

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

$h + 2 x$

Reorder $h$ and $2 x$ .

$2 x + h$

Step 5

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