Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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 $b = x$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Subtract $x$ from $x$ .

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

Step 1.2.2

Add $h$ and $0$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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Step 1.2.4.1

Apply the distributive property.

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

Step 1.2.4.2

Apply the distributive property.

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

Step 1.2.4.3

Apply the distributive property.

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

Step 1.2.5

Simplify and combine like terms.

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Step 1.2.5.1

Simplify each term.

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Step 1.2.5.1.1

Multiply $x$ by $x$ .

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

Step 1.2.5.1.2

Multiply $h$ by $h$ .

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

Step 1.2.5.2

Add $x h$ and $h x$ .

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Step 1.2.5.2.1

Reorder $x$ and $h$ .

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

Step 1.2.5.2.2

Add $h x$ and $h x$ .

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

Step 1.2.6

Apply the distributive property.

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

Step 1.2.7

Multiply $x$ by $x$ .

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $h$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Divide $3 x^{2} + h^{2} + 3 h x$ by $1$ .

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

Step 2.2

Move $3 x^{2}$ .

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