Calculus Examples

$f (x) = \frac{27 - 6 x - x^{2}}{x - 3}$

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.

$f (x + h) = \frac{27 - 6 (x + h) - {(x + h)}^{2}}{(x + h) - 3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Factor by grouping.

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Step 2.1.2.1.1.1

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 $a \cdot c = - 1 \cdot 27 = - 27$ and whose sum is $b = - 6$ .

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Step 2.1.2.1.1.1.1

Reorder terms.

$f (x + h) = \frac{- {(x + h)}^{2} + 27 - 6 (x + h)}{x + h - 3}$

Step 2.1.2.1.1.1.2

Reorder $27$ and $- 6 (x + h)$ .

$f (x + h) = \frac{- {(x + h)}^{2} - 6 (x + h) + 27}{x + h - 3}$

Step 2.1.2.1.1.1.3

Rewrite $- 6$ as $3$ plus $- 9$

$f (x + h) = \frac{- {(x + h)}^{2} + (3 - 9) (x + h) + 27}{x + h - 3}$

Step 2.1.2.1.1.1.4

Apply the distributive property.

$f (x + h) = \frac{- {(x + h)}^{2} + 3 (x + h) - 9 (x + h) + 27}{x + h - 3}$

Step 2.1.2.1.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.1.2.1

Group the first two terms and the last two terms.

$f (x + h) = \frac{(- {(x + h)}^{2} + 3 (x + h)) - 9 (x + h) + 27}{x + h - 3}$

Step 2.1.2.1.1.2.2

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

$f (x + h) = \frac{(x + h) (- (x + h) + 3) + 9 (- (x + h) + 3)}{x + h - 3}$

Step 2.1.2.1.1.3

Factor the polynomial by factoring out the greatest common factor, $- (x + h) + 3$ .

$f (x + h) = \frac{(- (x + h) + 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.1.2

Apply the distributive property.

$f (x + h) = \frac{(- x - h + 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2

Simplify terms.

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Step 2.1.2.2.1

Cancel the common factor of $- x - h + 3$ and $x + h - 3$ .

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Step 2.1.2.2.1.1

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

$f (x + h) = \frac{(- (x) - h + 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.2

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

$f (x + h) = \frac{(- (x) - (h) + 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.3

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

$f (x + h) = \frac{(- (x + h) + 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.4

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

$f (x + h) = \frac{(- (x + h) - 1 \cdot - 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.5

Factor $- 1$ out of $- (x + h) - 1 (- 3)$ .

$f (x + h) = \frac{- (x + h - 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.6

Rewrite $- (x + h - 3)$ as $- 1 (x + h - 3)$ .

$f (x + h) = \frac{- 1 (x + h - 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.7

Cancel the common factor.

$f (x + h) = \frac{- 1 (x + h - 3) (x + h + 9)}{x + h - 3}$

Step 2.1.2.2.1.8

Divide $- 1 (x + h + 9)$ by $1$ .

$f (x + h) = - 1 (x + h + 9)$

Step 2.1.2.2.2

Rewrite $- 1 (x + h + 9)$ as $- (x + h + 9)$ .

$f (x + h) = - (x + h + 9)$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = - x - h - 1 \cdot 9$

Step 2.1.2.2.4

Multiply $- 1$ by $9$ .

$f (x + h) = - x - h - 9$

Step 2.1.2.3

The final answer is $- x - h - 9$ .

$- x - h - 9$

Step 2.2

Reorder $- x$ and $- h$ .

$- h - x - 9$

Step 2.3

Find the components of the definition.

$f (x + h) = - h - x - 9$

$f (x) = - 9 - x$

$f (x + h) = - h - x - 9$

$f (x) = - 9 - x$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{- h - x - 9 - (- 9 - x)}{h}$

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.

$f' (x) = lim_{h \to 0} \frac{- h - x - 9 + 9 + x}{h}$

Step 4.1.2

Multiply $- 1$ by $- 9$ .

$f' (x) = lim_{h \to 0} \frac{- h - x - 9 + 9 + x}{h}$

Step 4.1.3

Multiply $- - x$ .

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{- h - x - 9 + 9 + 1 x}{h}$

Step 4.1.3.2

Multiply $x$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{- h - x - 9 + 9 + x}{h}$

Step 4.1.4

Add $- x$ and $x$ .

$f' (x) = lim_{h \to 0} \frac{- h + 0 - 9 + 9}{h}$

Step 4.1.5

Add $- h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{- h - 9 + 9}{h}$

Step 4.1.6

Add $- 9$ and $9$ .

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

Step 4.1.7

Add $- h$ and $0$ .

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

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $- 1$ by $1$ .

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

Step 5

Evaluate the limit of $- 1$ which is constant as $h$ approaches $0$ .

$- 1$

Step 6