Calculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.3.1.2

Multiply $h$ by $h$ .

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

Step 1.3.2

Add $x h$ and $h x$ .

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Step 1.3.2.1

Reorder $x$ and $h$ .

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

Step 1.3.2.2

Add $h x$ and $h x$ .

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Multiply $2$ by $3$ .

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

Step 1.6

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

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

Step 1.7

Subtract $x$ from $- x$ .

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

Step 1.8

Add $2$ and $2$ .

$\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x + h + 4}{h}$

Step 2

Split the fraction $\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x + h + 4}{h}$ into two fractions.

$\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x + h}{h} + \frac{4}{h}$

Step 3

Split the fraction $\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x + h}{h}$ into two fractions.

$\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 4

Split the fraction $\frac{6 x^{2} + 6 h x + 3 h^{2} - 2 x}{h}$ into two fractions.

$\frac{6 x^{2} + 6 h x + 3 h^{2}}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 5

Split the fraction $\frac{6 x^{2} + 6 h x + 3 h^{2}}{h}$ into two fractions.

$\frac{6 x^{2} + 6 h x}{h} + \frac{3 h^{2}}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 6

Split the fraction $\frac{6 x^{2} + 6 h x}{h}$ into two fractions.

$\frac{6 x^{2}}{h} + \frac{6 h x}{h} + \frac{3 h^{2}}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 7

Cancel the common factor of $h$ .

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Step 7.1

Cancel the common factor.

$\frac{6 x^{2}}{h} + \frac{6 h x}{h} + \frac{3 h^{2}}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 7.2

Divide $6 x$ by $1$ .

$\frac{6 x^{2}}{h} + 6 x + \frac{3 h^{2}}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8

Cancel the common factor of $h^{2}$ and $h$ .

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Step 8.1

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

$\frac{6 x^{2}}{h} + 6 x + \frac{h (3 h)}{h} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Raise $h$ to the power of $1$ .

$\frac{6 x^{2}}{h} + 6 x + \frac{h (3 h)}{h^{1}} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8.2.2

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

$\frac{6 x^{2}}{h} + 6 x + \frac{h (3 h)}{h \cdot 1} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8.2.3

Cancel the common factor.

$\frac{6 x^{2}}{h} + 6 x + \frac{h (3 h)}{h \cdot 1} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8.2.4

Rewrite the expression.

$\frac{6 x^{2}}{h} + 6 x + \frac{3 h}{1} + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 8.2.5

Divide $3 h$ by $1$ .

$\frac{6 x^{2}}{h} + 6 x + 3 h + \frac{- 2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 9

Move the negative in front of the fraction.

$\frac{6 x^{2}}{h} + 6 x + 3 h - \frac{2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 10

Cancel the common factor of $h$ .

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Step 10.1

Cancel the common factor.

$\frac{6 x^{2}}{h} + 6 x + 3 h - \frac{2 x}{h} + \frac{h}{h} + \frac{4}{h}$

Step 10.2

Rewrite the expression.

$\frac{6 x^{2}}{h} + 6 x + 3 h - \frac{2 x}{h} + 1 + \frac{4}{h}$