Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Multiply $h$ by $h$ .

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

Step 1.3.2

Add $x h$ and $h x$ .

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Step 1.3.2.1

Reorder $x$ and $h$ .

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

Step 1.3.2.2

Add $h x$ and $h x$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Multiply $2$ by $- 3$ .

$\frac{- 3 x^{2} - 6 (h x) - 3 h^{2} - 2 (x + h) - 2 - (- 3 x^{2} - 2 x - 2)}{h}$

Step 1.6

Apply the distributive property.

$\frac{- 3 x^{2} - 6 h x - 3 h^{2} - 2 x - 2 h - 2 - (- 3 x^{2} - 2 x - 2)}{h}$

Step 1.7

Apply the distributive property.

$\frac{- 3 x^{2} - 6 h x - 3 h^{2} - 2 x - 2 h - 2 - (- 3 x^{2}) - (- 2 x) - - 2}{h}$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $- 3$ by $- 1$ .

$\frac{- 3 x^{2} - 6 h x - 3 h^{2} - 2 x - 2 h - 2 + 3 x^{2} - (- 2 x) - - 2}{h}$

Step 1.8.2

Multiply $- 2$ by $- 1$ .

$\frac{- 3 x^{2} - 6 h x - 3 h^{2} - 2 x - 2 h - 2 + 3 x^{2} + 2 x - - 2}{h}$

Step 1.8.3

Multiply $- 1$ by $- 2$ .

$\frac{- 3 x^{2} - 6 h x - 3 h^{2} - 2 x - 2 h - 2 + 3 x^{2} + 2 x + 2}{h}$

Step 1.9

Add $- 3 x^{2}$ and $3 x^{2}$ .

$\frac{- 6 h x - 3 h^{2} - 2 x - 2 h - 2 + 0 + 2 x + 2}{h}$

Step 1.10

Add $- 6 h x$ and $0$ .

$\frac{- 6 h x - 3 h^{2} - 2 x - 2 h - 2 + 2 x + 2}{h}$

Step 1.11

Add $- 2 x$ and $2 x$ .

$\frac{- 6 h x - 3 h^{2} - 2 h - 2 + 0 + 2}{h}$

Step 1.12

Add $- 6 h x$ and $0$ .

$\frac{- 6 h x - 3 h^{2} - 2 h - 2 + 2}{h}$

Step 1.13

Add $- 2$ and $2$ .

$\frac{- 6 h x - 3 h^{2} - 2 h + 0}{h}$

Step 1.14

Add $- 6 h x - 3 h^{2} - 2 h$ and $0$ .

$\frac{- 6 h x - 3 h^{2} - 2 h}{h}$

Step 1.15

Factor $h$ out of $- 6 h x - 3 h^{2} - 2 h$ .

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Step 1.15.1

Factor $h$ out of $- 6 h x$ .

$\frac{h (- 6 x) - 3 h^{2} - 2 h}{h}$

Step 1.15.2

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

$\frac{h (- 6 x) + h (- 3 h) - 2 h}{h}$

Step 1.15.3

Factor $h$ out of $- 2 h$ .

$\frac{h (- 6 x) + h (- 3 h) + h \cdot - 2}{h}$

Step 1.15.4

Factor $h$ out of $h (- 6 x) + h (- 3 h)$ .

$\frac{h (- 6 x - 3 h) + h \cdot - 2}{h}$

Step 1.15.5

Factor $h$ out of $h (- 6 x - 3 h) + h \cdot - 2$ .

$\frac{h (- 6 x - 3 h - 2)}{h}$

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.

$\frac{h (- 6 x - 3 h - 2)}{h}$

Step 2.1.2

Divide $- 6 x - 3 h - 2$ by $1$ .

$- 6 x - 3 h - 2$

Step 2.2

Reorder $- 6 x$ and $- 3 h$ .

$- 3 h - 6 x - 2$