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

Simplify each term.

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Step 1.3.1.1

Use the Binomial Theorem.

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

Step 1.3.1.2

Apply the distributive property.

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

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

Multiply $3$ by $- 1$ .

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

Step 1.3.1.3.2

Multiply $3$ by $- 1$ .

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

Step 1.3.1.4

Remove parentheses.

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

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

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

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

Step 1.3.2.2

Move $- h^{3}$ .

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

Step 1.3.2.3

Move $- 3 a h^{2}$ .

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

Step 1.3.2.4

Move $- 3 a^{2} h$ .

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

Step 1.3.2.5

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

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

Step 2

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

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