Precalculus Examples

$f (x) = 14 x^{2}$

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.

$f (x + h) = 14 {(x + h)}^{2}$

Simplify the result.

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

$f (x + h) = 14 ((x + h) (x + h))$

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

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Apply the distributive property.

$f (x + h) = 14 (x (x + h) + h (x + h))$

Apply the distributive property.

$f (x + h) = 14 (x \cdot x + x h + h (x + h))$

Apply the distributive property.

$f (x + h) = 14 (x \cdot x + x h + h x + h \cdot h)$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$f (x + h) = 14 (x^{2} + x h + h x + h \cdot h)$

Multiply $h$ by $h$ .

$f (x + h) = 14 (x^{2} + x h + h x + h^{2})$

Add $x h$ and $h x$ .

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Reorder $x$ and $h$ .

$f (x + h) = 14 (x^{2} + h x + h x + h^{2})$

Add $h x$ and $h x$ .

$f (x + h) = 14 (x^{2} + 2 h x + h^{2})$

Apply the distributive property.

$f (x + h) = 14 x^{2} + 14 (2 h x) + 14 h^{2}$

Multiply $2$ by $14$ .

$f (x + h) = 14 x^{2} + 28 h x + 14 h^{2}$

The final answer is $14 x^{2} + 28 h x + 14 h^{2}$ .

$14 x^{2} + 28 h x + 14 h^{2}$

Find the components of the definition.

$f (x + h) = 14 x^{2} + 28 h x + 14 h^{2}$

$f (x) = 14 x^{2}$

$f (x + h) = 14 x^{2} + 28 h x + 14 h^{2}$

$f (x) = 14 x^{2}$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{14 x^{2} + 28 h x + 14 h^{2} - (14 x^{2})}{h}$

Simplify.

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Simplify the numerator.

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Multiply $- 1$ by $14$ .

$\frac{14 x^{2} + 28 h x + 14 h^{2} - 14 x^{2}}{h}$

Subtract $14 x^{2}$ from $14 x^{2}$ .

$\frac{28 h x + 14 h^{2} + 0}{h}$

Add $28 h x + 14 h^{2}$ and $0$ .

$\frac{28 h x + 14 h^{2}}{h}$

Factor $14 h$ out of $28 h x + 14 h^{2}$ .

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Factor $14 h$ out of $28 h x$ .

$\frac{14 h (2 x) + 14 h^{2}}{h}$

Factor $14 h$ out of $14 h^{2}$ .

$\frac{14 h (2 x) + 14 h \cdot h}{h}$

Factor $14 h$ out of $14 h (2 x) + 14 h \cdot h$ .

$\frac{14 h (2 x + h)}{h}$

Simplify terms.

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Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$\frac{14 h (2 x + h)}{h}$

Divide $14 (2 x + h)$ by $1$ .

$14 (2 x + h)$

Apply the distributive property.

$14 (2 x) + 14 h$

Multiply $2$ by $14$ .

$28 x + 14 h$

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