Precalculus Examples

$f (x) = x^{3}$ , $\frac{f (a + h) - f (a)}{h}$

Step 1

Evaluate the numerator in $\frac{f (a + h) - f a}{h}$ .

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Step 1.1

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

$f (a + h) = {(a + h)}^{3}$

Step 1.2

Replace the function designators in $f (a + h) - f a$ with the actual functions.

$({(a + h)}^{3}) - x^{3} a$

Step 1.3

Simplify $({(a + h)}^{3}) - x^{3} a$ .

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Step 1.3.1

Use the Binomial Theorem.

$a^{3} + 3 a^{2} h + 3 a h^{2} + h^{3} - x^{3} a$

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

Move $x^{3}$ .

$a^{3} + 3 a^{2} h + 3 a h^{2} + h^{3} - 1 a x^{3}$

Step 1.3.2.2

Move $h^{3}$ .

$a^{3} + 3 a^{2} h + 3 a h^{2} - 1 a x^{3} + h^{3}$

Step 1.3.2.3

Move $3 a h^{2}$ .

$a^{3} + 3 a^{2} h - 1 a x^{3} + 3 a h^{2} + h^{3}$

Step 1.3.2.4

Move $3 a^{2} h$ .

$a^{3} - 1 a x^{3} + 3 a^{2} h + 3 a h^{2} + h^{3}$

Step 1.3.2.5

Reorder $a^{3}$ and $- 1 a x^{3}$ .

$- 1 a x^{3} + a^{3} + 3 a^{2} h + 3 a h^{2} + h^{3}$

Step 2

Rewrite $\frac{f (a + h) - f a}{h}$ .

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

Step 3

Simplify $\frac{- 1 a x^{3} + a^{3} + 3 a^{2} h + 3 a h^{2} + h^{3}}{h}$ .

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Step 3.1

Factor $- 1$ out of $- 1 a x^{3}$ .

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

Step 3.2

Factor $- 1$ out of $a^{3}$ .

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

Step 3.3

Factor $- 1$ out of $- (a x^{3}) - 1 (- a^{3})$ .

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

Step 3.4

Factor $- 1$ out of $3 a^{2} h$ .

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

Step 3.5

Factor $- 1$ out of $- (a x^{3} - a^{3}) - (- 3 a^{2} h)$ .

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

Step 3.6

Factor $- 1$ out of $3 a h^{2}$ .

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

Step 3.7

Factor $- 1$ out of $- (a x^{3} - a^{3} - 3 a^{2} h) - (- 3 a h^{2})$ .

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

Step 3.8

Factor $- 1$ out of $h^{3}$ .

$\frac{- (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2}) - 1 (- h^{3})}{h}$

Step 3.9

Factor $- 1$ out of $- (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2}) - 1 (- h^{3})$ .

$\frac{- (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2} - h^{3})}{h}$

Step 3.10

Simplify the expression.

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Step 3.10.1

Rewrite $- (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2} - h^{3})$ as $- 1 (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2} - h^{3})$ .

$\frac{- 1 (a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2} - h^{3})}{h}$

Step 3.10.2

Move the negative in front of the fraction.

$- \frac{a x^{3} - a^{3} - 3 a^{2} h - 3 a h^{2} - h^{3}}{h}$