Calculus Examples

$f (x) = \frac{1}{3} x^{3} - x^{2} - 4 x + \frac{32}{3}$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{1}{3} \cdot {(x + h)}^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Use the Binomial Theorem.

$f (x + h) = \frac{1}{3} \cdot (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.2

Apply the distributive property.

$f (x + h) = \frac{1}{3} x^{3} + \frac{1}{3} \cdot (3 x^{2} h) + \frac{1}{3} \cdot (3 x h^{2}) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3

Simplify.

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Step 2.1.2.1.3.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$f (x + h) = \frac{x^{3}}{3} + \frac{1}{3} \cdot (3 x^{2} h) + \frac{1}{3} \cdot (3 x h^{2}) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.1.2.1.3.2.1

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

$f (x + h) = \frac{x^{3}}{3} + \frac{1}{3} \cdot (3 (x^{2} h)) + \frac{1}{3} \cdot (3 x h^{2}) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.2.2

Cancel the common factor.

$f (x + h) = \frac{x^{3}}{3} + \frac{1}{3} \cdot (3 (x^{2} h)) + \frac{1}{3} \cdot (3 x h^{2}) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.2.3

Rewrite the expression.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + \frac{1}{3} \cdot (3 x h^{2}) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.3

Cancel the common factor of $3$ .

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Step 2.1.2.1.3.3.1

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

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + \frac{1}{3} \cdot (3 (x h^{2})) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.3.2

Cancel the common factor.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + \frac{1}{3} \cdot (3 (x h^{2})) + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.3.3

Rewrite the expression.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{1}{3} h^{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.3.4

Combine $\frac{1}{3}$ and $h^{3}$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - {(x + h)}^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.4

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

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - ((x + h) (x + h)) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.5

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

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Step 2.1.2.1.5.1

Apply the distributive property.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x (x + h) + h (x + h)) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.5.2

Apply the distributive property.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x \cdot x + x h + h (x + h)) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.5.3

Apply the distributive property.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x \cdot x + x h + h x + h \cdot h) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.6

Simplify and combine like terms.

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Step 2.1.2.1.6.1

Simplify each term.

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Step 2.1.2.1.6.1.1

Multiply $x$ by $x$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x^{2} + x h + h x + h \cdot h) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.6.1.2

Multiply $h$ by $h$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x^{2} + x h + h x + h^{2}) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.6.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.6.2.1

Reorder $x$ and $h$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x^{2} + h x + h x + h^{2}) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.6.2.2

Add $h x$ and $h x$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - (x^{2} + 2 h x + h^{2}) - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.7

Apply the distributive property.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - x^{2} - (2 h x) - h^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.8

Multiply $2$ by $- 1$ .

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - x^{2} - 2 (h x) - h^{2} - 4 (x + h) + \frac{32}{3}$

Step 2.1.2.1.9

Apply the distributive property.

$f (x + h) = \frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 x - 4 h + \frac{32}{3}$

Step 2.1.2.2

The final answer is $\frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 x - 4 h + \frac{32}{3}$ .

$\frac{x^{3}}{3} + x^{2} h + x h^{2} + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 x - 4 h + \frac{32}{3}$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x^{2}$ and $h$ .

$\frac{x^{3}}{3} + h x^{2} + x h^{2} + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 x - 4 h + \frac{32}{3}$

Step 2.2.2

Reorder $x$ and $h^{2}$ .

$\frac{x^{3}}{3} + h x^{2} + h^{2} x + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 x - 4 h + \frac{32}{3}$

Step 2.2.3

Move $- 4 x$ .

$\frac{x^{3}}{3} + h x^{2} + h^{2} x + \frac{h^{3}}{3} - x^{2} - 2 h x - h^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.2.4

Move $- x^{2}$ .

$\frac{x^{3}}{3} + h x^{2} + h^{2} x + \frac{h^{3}}{3} - 2 h x - h^{2} - x^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.2.5

Move $- 2 h x$ .

$\frac{x^{3}}{3} + h x^{2} + h^{2} x + \frac{h^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.2.6

Move $\frac{x^{3}}{3}$ .

$h x^{2} + h^{2} x + \frac{h^{3}}{3} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.2.7

Move $h x^{2}$ .

$h^{2} x + \frac{h^{3}}{3} + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.2.8

Reorder $h^{2} x$ and $\frac{h^{3}}{3}$ .

$\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

Step 2.3

Find the components of the definition.

$f (x + h) = \frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

$f (x) = \frac{x^{3}}{3} - x^{2} - 4 x + \frac{32}{3}$

$f (x + h) = \frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3}$

$f (x) = \frac{x^{3}}{3} - x^{2} - 4 x + \frac{32}{3}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} - (\frac{x^{3}}{3} - x^{2} - 4 x + \frac{32}{3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} - \frac{x^{3}}{3} + x^{2} - (- 4 x) - \frac{32}{3}}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- - x^{2}$ .

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Step 4.1.2.1.1

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} - \frac{x^{3}}{3} + 1 x^{2} - (- 4 x) - \frac{32}{3}}{h}$

Step 4.1.2.1.2

Multiply $x^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} - \frac{x^{3}}{3} + x^{2} - (- 4 x) - \frac{32}{3}}{h}$

Step 4.1.2.2

Multiply $- 4$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} + \frac{x^{3}}{3} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} - \frac{x^{3}}{3} + x^{2} + 4 x - \frac{32}{3}}{h}$

Step 4.1.3

Subtract $\frac{x^{3}}{3}$ from $\frac{x^{3}}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} + 0 + x^{2} + 4 x - \frac{32}{3}}{h}$

Step 4.1.4

Add $\frac{h^{3}}{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - x^{2} - 4 h - 4 x + \frac{32}{3} + x^{2} + 4 x - \frac{32}{3}}{h}$

Step 4.1.5

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

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h - 4 x + \frac{32}{3} + 0 + 4 x - \frac{32}{3}}{h}$

Step 4.1.6

Add $\frac{h^{3}}{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h - 4 x + \frac{32}{3} + 4 x - \frac{32}{3}}{h}$

Step 4.1.7

Add $- 4 x$ and $4 x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + 0 + \frac{32}{3} - \frac{32}{3}}{h}$

Step 4.1.8

Add $\frac{h^{3}}{3}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + \frac{32}{3} - \frac{32}{3}}{h}$

Step 4.1.9

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + \frac{32 - 32}{3}}{h}$

Step 4.1.10

Subtract $32$ from $32$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + \frac{0}{3}}{h}$

Step 4.1.11

Combine the opposite terms in $\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + \frac{0}{3}$ .

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Step 4.1.11.1

Divide $0$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h + 0}{h}$

Step 4.1.11.2

Add $\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.12

To write $h^{2} x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + h^{2} x \cdot \frac{3}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.13

Combine $h^{2} x$ and $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3}}{3} + \frac{h^{2} x \cdot 3}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.14

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h^{3} + h^{2} x \cdot 3}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.15

Simplify the numerator.

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Step 4.1.15.1

Factor $h^{2}$ out of $h^{3} + h^{2} x \cdot 3$ .

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Step 4.1.15.1.1

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

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} h + h^{2} x \cdot 3}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.15.1.2

Factor $h^{2}$ out of $h^{2} x \cdot 3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} h + h^{2} (x \cdot 3)}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.15.1.3

Factor $h^{2}$ out of $h^{2} h + h^{2} (x \cdot 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} (h + x \cdot 3)}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.15.2

Move $3$ to the left of $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} (h + 3 x)}{3} + h x^{2} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.16

To write $h x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} (h + 3 x)}{3} + h x^{2} \cdot \frac{3}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.17

Combine $h x^{2}$ and $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} (h + 3 x)}{3} + \frac{h x^{2} \cdot 3}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.18

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h^{2} (h + 3 x) + h x^{2} \cdot 3}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19

Simplify the numerator.

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Step 4.1.19.1

Factor $h$ out of $h^{2} (h + 3 x) + h x^{2} \cdot 3$ .

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Step 4.1.19.1.1

Factor $h$ out of $h^{2} (h + 3 x)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h (h + 3 x)) + h (x^{2}) \cdot 3}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.1.2

Factor $h$ out of $h x^{2} \cdot 3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h (h + 3 x)) + h (x^{2} \cdot 3)}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.1.3

Factor $h$ out of $h (h (h + 3 x)) + h (x^{2} \cdot 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h (h + 3 x) + x^{2} \cdot 3)}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{h (h (h) + h (3 x) + x^{2} \cdot 3)}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.3

Multiply $h$ by $h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + h (3 x) + x^{2} \cdot 3)}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.4

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + x^{2} \cdot 3)}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.19.5

Move $3$ to the left of $x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2})}{3} - h^{2} - 2 h x - 4 h}{h}$

Step 4.1.20

To write $- h^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2})}{3} - h^{2} \cdot \frac{3}{3} - 2 h x - 4 h}{h}$

Step 4.1.21

Combine $- h^{2}$ and $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2})}{3} + \frac{- h^{2} \cdot 3}{3} - 2 h x - 4 h}{h}$

Step 4.1.22

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2}) - h^{2} \cdot 3}{3} - 2 h x - 4 h}{h}$

Step 4.1.23

Simplify the numerator.

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Step 4.1.23.1

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2}) - h^{2} \cdot 3$ .

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Step 4.1.23.1.1

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

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2}) + h (- h \cdot 3)}{3} - 2 h x - 4 h}{h}$

Step 4.1.23.1.2

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2}) + h (- h \cdot 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - h \cdot 3)}{3} - 2 h x - 4 h}{h}$

Step 4.1.23.2

Multiply $3$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h)}{3} - 2 h x - 4 h}{h}$

Step 4.1.24

To write $- 2 h x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h)}{3} - 2 h x \cdot \frac{3}{3} - 4 h}{h}$

Step 4.1.25

Combine $- 2 h x$ and $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h)}{3} + \frac{- 2 h x \cdot 3}{3} - 4 h}{h}$

Step 4.1.26

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h) - 2 h x \cdot 3}{3} - 4 h}{h}$

Step 4.1.27

Simplify the numerator.

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Step 4.1.27.1

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 3 h) - 2 h x \cdot 3$ .

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Step 4.1.27.1.1

Factor $h$ out of $- 2 h x \cdot 3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h) + h (- 2 x \cdot 3)}{3} - 4 h}{h}$

Step 4.1.27.1.2

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 3 h) + h (- 2 x \cdot 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 2 x \cdot 3)}{3} - 4 h}{h}$

Step 4.1.27.2

Multiply $3$ by $- 2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x)}{3} - 4 h}{h}$

Step 4.1.28

To write $- 4 h$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x)}{3} - 4 h \cdot \frac{3}{3}}{h}$

Step 4.1.29

Combine $- 4 h$ and $\frac{3}{3}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x)}{3} + \frac{- 4 h \cdot 3}{3}}{h}$

Step 4.1.30

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) - 4 h \cdot 3}{3}}{h}$

Step 4.1.31

Simplify the numerator.

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Step 4.1.31.1

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) - 4 h \cdot 3$ .

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Step 4.1.31.1.1

Factor $h$ out of $- 4 h \cdot 3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) + h (- 4 \cdot 3)}{3}}{h}$

Step 4.1.31.1.2

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) + h (- 4 \cdot 3)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 4 \cdot 3)}{3}}{h}$

Step 4.1.31.2

Multiply $- 4$ by $3$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12)}{3}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12)}{3} \cdot \frac{1}{h}$

Step 4.3

Combine.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12) \cdot 1}{3 h}$

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12) \cdot 1}{3 h}$

Step 4.4.2

Rewrite the expression.

$f' (x) = lim_{h \to 0} \frac{(h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12) \cdot 1}{3}$

Step 4.5

Multiply $h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12}{3}$

Step 5

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{3} lim_{h \to 0} h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x - 12$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{1}{3} (lim_{h \to 0} h^{2} + lim_{h \to 0} 3 h x + lim_{h \to 0} 3 x^{2} - lim_{h \to 0} 3 h - lim_{h \to 0} 6 x - lim_{h \to 0} 12)$

Step 7

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + lim_{h \to 0} 3 h x + lim_{h \to 0} 3 x^{2} - lim_{h \to 0} 3 h - lim_{h \to 0} 6 x - lim_{h \to 0} 12)$

Step 8

Move the term $3 x$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + 3 x lim_{h \to 0} h + lim_{h \to 0} 3 x^{2} - lim_{h \to 0} 3 h - lim_{h \to 0} 6 x - lim_{h \to 0} 12)$

Step 9

Evaluate the limit of $3 x^{2}$ which is constant as $h$ approaches $0$ .

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + 3 x lim_{h \to 0} h + 3 x^{2} - lim_{h \to 0} 3 h - lim_{h \to 0} 6 x - lim_{h \to 0} 12)$

Step 10

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + 3 x lim_{h \to 0} h + 3 x^{2} - 3 lim_{h \to 0} h - lim_{h \to 0} 6 x - lim_{h \to 0} 12)$

Step 11

Evaluate the limit of $6 x$ which is constant as $h$ approaches $0$ .

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + 3 x lim_{h \to 0} h + 3 x^{2} - 3 lim_{h \to 0} h - 6 x - lim_{h \to 0} 12)$

Step 12

Evaluate the limit of $12$ which is constant as $h$ approaches $0$ .

$\frac{1}{3} ({(lim_{h \to 0} h)}^{2} + 3 x lim_{h \to 0} h + 3 x^{2} - 3 lim_{h \to 0} h - 6 x - 1 \cdot 12)$

Step 13

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 13.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{3} (0^{2} + 3 x lim_{h \to 0} h + 3 x^{2} - 3 lim_{h \to 0} h - 6 x - 1 \cdot 12)$

Step 13.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{3} (0^{2} + 3 x \cdot 0 + 3 x^{2} - 3 lim_{h \to 0} h - 6 x - 1 \cdot 12)$

Step 13.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{1}{3} (0^{2} + 3 x \cdot 0 + 3 x^{2} - 3 \cdot 0 - 6 x - 1 \cdot 12)$

Step 14

Simplify the answer.

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Step 14.1

Simplify each term.

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Step 14.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} (0 + 3 x \cdot 0 + 3 x^{2} - 3 \cdot 0 - 6 x - 1 \cdot 12)$

Step 14.1.2

Multiply $3 x \cdot 0$ .

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Step 14.1.2.1

Multiply $0$ by $3$ .

$\frac{1}{3} (0 + 0 x + 3 x^{2} - 3 \cdot 0 - 6 x - 1 \cdot 12)$

Step 14.1.2.2

Multiply $0$ by $x$ .

$\frac{1}{3} (0 + 0 + 3 x^{2} - 3 \cdot 0 - 6 x - 1 \cdot 12)$

Step 14.1.3

Multiply $- 3$ by $0$ .

$\frac{1}{3} (0 + 0 + 3 x^{2} + 0 - 6 x - 1 \cdot 12)$

Step 14.1.4

Multiply $- 1$ by $12$ .

$\frac{1}{3} (0 + 0 + 3 x^{2} + 0 - 6 x - 12)$

Step 14.2

Combine the opposite terms in $0 + 0 + 3 x^{2} + 0 - 6 x - 12$ .

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Step 14.2.1

Add $0$ and $0$ .

$\frac{1}{3} (0 + 3 x^{2} + 0 - 6 x - 12)$

Step 14.2.2

Add $0$ and $3 x^{2}$ .

$\frac{1}{3} (3 x^{2} + 0 - 6 x - 12)$

Step 14.2.3

Add $3 x^{2}$ and $0$ .

$\frac{1}{3} (3 x^{2} - 6 x - 12)$

Step 14.3

Apply the distributive property.

$\frac{1}{3} (3 x^{2}) + \frac{1}{3} (- 6 x) + \frac{1}{3} \cdot - 12$

Step 14.4

Simplify.

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Step 14.4.1

Cancel the common factor of $3$ .

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Step 14.4.1.1

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

$\frac{1}{3} (3 (x^{2})) + \frac{1}{3} (- 6 x) + \frac{1}{3} \cdot - 12$

Step 14.4.1.2

Cancel the common factor.

$\frac{1}{3} (3 x^{2}) + \frac{1}{3} (- 6 x) + \frac{1}{3} \cdot - 12$

Step 14.4.1.3

Rewrite the expression.

$x^{2} + \frac{1}{3} (- 6 x) + \frac{1}{3} \cdot - 12$

Step 14.4.2

Cancel the common factor of $3$ .

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Step 14.4.2.1

Factor $3$ out of $- 6 x$ .

$x^{2} + \frac{1}{3} (3 (- 2 x)) + \frac{1}{3} \cdot - 12$

Step 14.4.2.2

Cancel the common factor.

$x^{2} + \frac{1}{3} (3 (- 2 x)) + \frac{1}{3} \cdot - 12$

Step 14.4.2.3

Rewrite the expression.

$x^{2} - 2 x + \frac{1}{3} \cdot - 12$

Step 14.4.3

Cancel the common factor of $3$ .

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Step 14.4.3.1

Factor $3$ out of $- 12$ .

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

Step 14.4.3.2

Cancel the common factor.

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

Step 14.4.3.3

Rewrite the expression.

$x^{2} - 2 x - 4$

Step 15