Calculus Examples

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

Step 1

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x + h$ and $b = x$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Subtract $x$ from $x$ .

$(h + 0) ({(x + h)}^{2} + (x + h) x + x^{2})$

Step 2.2

Add $h$ and $0$ .

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

Step 2.3

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

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Apply the distributive property.

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

Step 2.5

Simplify and combine like terms.

Tap for more steps...

Step 2.5.1

Simplify each term.

Tap for more steps...

Step 2.5.1.1

Multiply $x$ by $x$ .

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

Step 2.5.1.2

Multiply $h$ by $h$ .

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

Step 2.5.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.5.2.1

Reorder $x$ and $h$ .

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

Step 2.5.2.2

Add $h x$ and $h x$ .

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

Step 2.6

Apply the distributive property.

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

Step 2.7

Multiply $x$ by $x$ .

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

Step 2.8

Add $x^{2}$ and $x^{2}$ .

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

Step 2.9

Add $2 x^{2}$ and $x^{2}$ .

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

Step 2.10

Add $2 h x$ and $h x$ .

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