Calculus Examples

$f (x) = \frac{4 x - 2 x^{2} + 6}{2 x}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Cancel the common factor of $4 (x + h) - 2 {(x + h)}^{2} + 6$ and $2$ .

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Step 2.1.2.1.1

Factor $2$ out of $4 (x + h)$ .

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

Step 2.1.2.1.2

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

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

Step 2.1.2.1.3

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

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

Step 2.1.2.1.4

Factor $2$ out of $6$ .

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

Step 2.1.2.1.5

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

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

Step 2.1.2.1.6

Cancel the common factors.

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Step 2.1.2.1.6.1

Cancel the common factor.

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

Step 2.1.2.1.6.2

Rewrite the expression.

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

Step 2.1.2.2

Simplify the numerator.

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Step 2.1.2.2.1

Let $u = x + h$ . Substitute $u$ for all occurrences of $x + h$ .

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

Step 2.1.2.2.2

Factor by grouping.

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Step 2.1.2.2.2.1

Reorder terms.

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

Step 2.1.2.2.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = 2$ .

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Step 2.1.2.2.2.2.1

Factor $2$ out of $2 u$ .

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

Step 2.1.2.2.2.2.2

Rewrite $2$ as $- 1$ plus $3$

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

Step 2.1.2.2.2.2.3

Apply the distributive property.

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

Step 2.1.2.2.2.3

Factor out the greatest common factor from each group.

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Step 2.1.2.2.2.3.1

Group the first two terms and the last two terms.

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

Step 2.1.2.2.2.3.2

Factor out the greatest common factor (GCF) from each group.

$f (x + h) = \frac{u (- u - 1) - 3 (- u - 1)}{x + h}$

Step 2.1.2.2.2.4

Factor the polynomial by factoring out the greatest common factor, $- u - 1$ .

$f (x + h) = \frac{(- u - 1) (u - 3)}{x + h}$

Step 2.1.2.2.3

Replace all occurrences of $u$ with $x + h$ .

$f (x + h) = \frac{(- (x + h) - 1) (x + h - 3)}{x + h}$

Step 2.1.2.2.4

Apply the distributive property.

$f (x + h) = \frac{(- x - h - 1) (x + h - 3)}{x + h}$

Step 2.1.2.3

Simplify with factoring out.

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Step 2.1.2.3.1

Factor $- 1$ out of $- x$ .

$f (x + h) = \frac{(- (x) - h - 1) (x + h - 3)}{x + h}$

Step 2.1.2.3.2

Factor $- 1$ out of $- h$ .

$f (x + h) = \frac{(- (x) - (h) - 1) (x + h - 3)}{x + h}$

Step 2.1.2.3.3

Factor $- 1$ out of $- (x) - (h)$ .

$f (x + h) = \frac{(- (x + h) - 1) (x + h - 3)}{x + h}$

Step 2.1.2.3.4

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.1.2.3.5

Factor $- 1$ out of $- (x + h) - 1 (1)$ .

$f (x + h) = \frac{- (x + h + 1) (x + h - 3)}{x + h}$

Step 2.1.2.3.6

Simplify the expression.

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Step 2.1.2.3.6.1

Rewrite $- (x + h + 1)$ as $- 1 (x + h + 1)$ .

$f (x + h) = \frac{- 1 (x + h + 1) (x + h - 3)}{x + h}$

Step 2.1.2.3.6.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{(x + h + 1) (x + h - 3)}{x + h}$

Step 2.1.2.4

The final answer is $- \frac{(x + h + 1) (x + h - 3)}{x + h}$ .

$- \frac{(x + h + 1) (x + h - 3)}{x + h}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{(x + h + 1) (x + h - 3)}{x + h}$

$f (x) = - \frac{(x + 1) (x - 3)}{x}$

$f (x + h) = - \frac{(x + h + 1) (x + h - 3)}{x + h}$

$f (x) = - \frac{(x + 1) (x - 3)}{x}$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{(x + 1) (x - 3)}{x}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.1.2

Multiply $\frac{(x + 1) (x - 3)}{x}$ by $1$ .

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

Step 4.1.2

To write $- \frac{(x + h + 1) (x + h - 3)}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.1.3

To write $\frac{(x + 1) (x - 3)}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

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

Step 4.1.4

Write each expression with a common denominator of $(x + h) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1

Multiply $\frac{(x + h + 1) (x + h - 3)}{x + h}$ by $\frac{x}{x}$ .

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

Step 4.1.4.2

Multiply $\frac{(x + 1) (x - 3)}{x}$ by $\frac{x + h}{x + h}$ .

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

Step 4.1.4.3

Reorder the factors of $(x + h) x$ .

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

Step 4.1.5

Combine the numerators over the common denominator.

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

Step 4.1.6

Rewrite $\frac{- (x + h + 1) (x + h - 3) x + (x + 1) (x - 3) (x + h)}{x (x + h)}$ in a factored form.

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Step 4.1.6.1

Apply the distributive property.

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

Step 4.1.6.2

Multiply $- 1$ by $1$ .

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

Step 4.1.6.3

Expand $(- x - h - 1) (x + h - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.6.4

Simplify each term.

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Step 4.1.6.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.6.4.1.1

Move $x$ .

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

Step 4.1.6.4.1.2

Multiply $x$ by $x$ .

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

Step 4.1.6.4.2

Multiply $- 3$ by $- 1$ .

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

Step 4.1.6.4.3

Multiply $h$ by $h$ by adding the exponents.

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Step 4.1.6.4.3.1

Move $h$ .

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

Step 4.1.6.4.3.2

Multiply $h$ by $h$ .

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

Step 4.1.6.4.4

Multiply $- 3$ by $- 1$ .

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

Step 4.1.6.4.5

Rewrite $- 1 x$ as $- x$ .

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

Step 4.1.6.4.6

Rewrite $- 1 h$ as $- h$ .

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

Step 4.1.6.4.7

Multiply $- 1$ by $- 3$ .

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

Step 4.1.6.5

Subtract $h x$ from $- x h$ .

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Step 4.1.6.5.1

Move $h$ .

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

Step 4.1.6.5.2

Subtract $x h$ from $- x h$ .

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

Step 4.1.6.6

Subtract $x$ from $3 x$ .

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

Step 4.1.6.7

Subtract $h$ from $3 h$ .

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

Step 4.1.6.8

Apply the distributive property.

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

Step 4.1.6.9

Simplify.

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Step 4.1.6.9.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.6.9.1.1

Move $x$ .

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

Step 4.1.6.9.1.2

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

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Step 4.1.6.9.1.2.1

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

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

Step 4.1.6.9.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.6.9.1.3

Add $1$ and $2$ .

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

Step 4.1.6.9.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.6.9.2.1

Move $x$ .

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

Step 4.1.6.9.2.2

Multiply $x$ by $x$ .

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

Step 4.1.6.9.3

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.6.9.3.1

Move $x$ .

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

Step 4.1.6.9.3.2

Multiply $x$ by $x$ .

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

Step 4.1.6.10

Expand $(x + 1) (x - 3)$ using the FOIL Method.

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Step 4.1.6.10.1

Apply the distributive property.

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

Step 4.1.6.10.2

Apply the distributive property.

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

Step 4.1.6.10.3

Apply the distributive property.

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

Step 4.1.6.11

Simplify and combine like terms.

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Step 4.1.6.11.1

Simplify each term.

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Step 4.1.6.11.1.1

Multiply $x$ by $x$ .

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

Step 4.1.6.11.1.2

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

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

Step 4.1.6.11.1.3

Multiply $x$ by $1$ .

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

Step 4.1.6.11.1.4

Multiply $- 3$ by $1$ .

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

Step 4.1.6.11.2

Add $- 3 x$ and $x$ .

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

Step 4.1.6.12

Expand $(x^{2} - 2 x - 3) (x + h)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.6.13

Simplify each term.

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Step 4.1.6.13.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.6.13.1.1

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

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Step 4.1.6.13.1.1.1

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

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

Step 4.1.6.13.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.6.13.1.2

Add $2$ and $1$ .

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

Step 4.1.6.13.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.6.13.2.1

Move $x$ .

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

Step 4.1.6.13.2.2

Multiply $x$ by $x$ .

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

Step 4.1.6.14

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

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

Step 4.1.6.15

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

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

Step 4.1.6.16

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

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

Step 4.1.6.17

Subtract $2 x^{2}$ from $2 x^{2}$ .

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

Step 4.1.6.18

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

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

Step 4.1.6.19

Subtract $2 x h$ from $2 h x$ .

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Step 4.1.6.19.1

Move $h$ .

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

Step 4.1.6.19.2

Subtract $2 x h$ from $2 x h$ .

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

Step 4.1.6.20

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

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

Step 4.1.6.21

Subtract $3 x$ from $3 x$ .

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

Step 4.1.6.22

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

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

Step 4.1.6.23

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

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Step 4.1.6.23.1

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

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

Step 4.1.6.23.2

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

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

Step 4.1.6.23.3

Factor $h$ out of $- 3 h$ .

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

Step 4.1.6.23.4

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

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

Step 4.1.6.23.5

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

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Combine.

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

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

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

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

Step 4.6

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.7

Factor $- 1$ out of $- h x$ .

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

Step 4.8

Factor $- 1$ out of $- (x^{2}) - (h x)$ .

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

Step 4.9

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.10

Factor $- 1$ out of $- (x^{2} + h x) - 1 (3)$ .

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

Step 4.11

Simplify the expression.

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Step 4.11.1

Rewrite $- (x^{2} + h x + 3)$ as $- 1 (x^{2} + h x + 3)$ .

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

Step 4.11.2

Move the negative in front of the fraction.

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

Step 5

Move the term $- 1$ outside of the limit because it is constant with respect to $h$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

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

Step 12

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

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

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

$- \frac{1}{x} \cdot \frac{x^{2} + x \cdot 0 + 3}{x + 0}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Multiply $x$ by $0$ .

$- \frac{1}{x} \cdot \frac{x^{2} + 0 + 3}{x + 0}$

Step 15.1.2

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

$- \frac{1}{x} \cdot \frac{x^{2} + 3}{x + 0}$

Step 15.2

Add $x$ and $0$ .

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

Step 15.3

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

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Step 15.3.1

Multiply $\frac{x^{2} + 3}{x}$ by $\frac{1}{x}$ .

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

Step 15.3.2

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

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

Step 15.3.3

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

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

Step 15.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 15.3.5

Add $1$ and $1$ .

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

Step 16