Calculus Examples

$f (x) = \frac{- x - 6}{{(x - 6)}^{2}} + 9$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $\frac{- x - 6}{{(x - 6)}^{2}} + 9$ with respect to $x$ is $\frac{d}{d x} [\frac{- x - 6}{{(x - 6)}^{2}}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [\frac{- x - 6}{{(x - 6)}^{2}}] + \frac{d}{d x} [9]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{- x - 6}{{(x - 6)}^{2}}]$ .

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Step 1.1.2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = - x - 6$ and $g (x) = {(x - 6)}^{2}$ .

$\frac{{(x - 6)}^{2} \frac{d}{d x} [- x - 6] - (- x - 6) \frac{d}{d x} [{(x - 6)}^{2}]}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.2

By the Sum Rule, the derivative of $- x - 6$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [- 6]$ .

$\frac{{(x - 6)}^{2} (\frac{d}{d x} [- x] + \frac{d}{d x} [- 6]) - (- x - 6) \frac{d}{d x} [{(x - 6)}^{2}]}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{{(x - 6)}^{2} (- \frac{d}{d x} [x] + \frac{d}{d x} [- 6]) - (- x - 6) \frac{d}{d x} [{(x - 6)}^{2}]}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + \frac{d}{d x} [- 6]) - (- x - 6) \frac{d}{d x} [{(x - 6)}^{2}]}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) \frac{d}{d x} [{(x - 6)}^{2}]}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 6$ .

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Step 1.1.2.6.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 6])}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (2 u \frac{d}{d x} [x - 6])}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.6.3

Replace all occurrences of $u$ with $x - 6$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (2 (x - 6) \frac{d}{d x} [x - 6])}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.7

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (2 (x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (2 (x - 6) (1 + \frac{d}{d x} [- 6]))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.9

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{{(x - 6)}^{2} (- 1 \cdot 1 + 0) - (- x - 6) (2 (x - 6) (1 + 0))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.10

Multiply $- 1$ by $1$ .

$\frac{{(x - 6)}^{2} (- 1 + 0) - (- x - 6) (2 (x - 6) (1 + 0))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.11

Add $- 1$ and $0$ .

$\frac{{(x - 6)}^{2} \cdot - 1 - (- x - 6) (2 (x - 6) (1 + 0))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.12

Move $- 1$ to the left of ${(x - 6)}^{2}$ .

$\frac{- 1 \cdot {(x - 6)}^{2} - (- x - 6) (2 (x - 6) (1 + 0))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.13

Rewrite $- 1 {(x - 6)}^{2}$ as $- {(x - 6)}^{2}$ .

$\frac{- {(x - 6)}^{2} - (- x - 6) (2 (x - 6) (1 + 0))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.14

Add $1$ and $0$ .

$\frac{- {(x - 6)}^{2} - (- x - 6) (2 (x - 6) \cdot 1)}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.15

Multiply $2$ by $1$ .

$\frac{- {(x - 6)}^{2} - (- x - 6) (2 (x - 6))}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.16

Multiply $2$ by $- 1$ .

$\frac{- {(x - 6)}^{2} - 2 (- x - 6) (x - 6)}{{({(x - 6)}^{2})}^{2}} + \frac{d}{d x} [9]$

Step 1.1.2.17

Multiply the exponents in ${({(x - 6)}^{2})}^{2}$ .

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Step 1.1.2.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- {(x - 6)}^{2} - 2 (- x - 6) (x - 6)}{{(x - 6)}^{2 \cdot 2}} + \frac{d}{d x} [9]$

Step 1.1.2.17.2

Multiply $2$ by $2$ .

$\frac{- {(x - 6)}^{2} - 2 (- x - 6) (x - 6)}{{(x - 6)}^{4}} + \frac{d}{d x} [9]$

Step 1.1.2.18

Factor $x - 6$ out of $- {(x - 6)}^{2} - 2 (- x - 6) (x - 6)$ .

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Step 1.1.2.18.1

Factor $x - 6$ out of $- {(x - 6)}^{2}$ .

$\frac{(x - 6) (- (x - 6)) - 2 (- x - 6) (x - 6)}{{(x - 6)}^{4}} + \frac{d}{d x} [9]$

Step 1.1.2.18.2

Factor $x - 6$ out of $- 2 (- x - 6) (x - 6)$ .

$\frac{(x - 6) (- (x - 6)) + (x - 6) (- 2 (- x - 6))}{{(x - 6)}^{4}} + \frac{d}{d x} [9]$

Step 1.1.2.18.3

Factor $x - 6$ out of $(x - 6) (- (x - 6)) + (x - 6) (- 2 (- x - 6))$ .

$\frac{(x - 6) (- (x - 6) - 2 (- x - 6))}{{(x - 6)}^{4}} + \frac{d}{d x} [9]$

Step 1.1.2.19

Cancel the common factors.

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Step 1.1.2.19.1

Factor $x - 6$ out of ${(x - 6)}^{4}$ .

$\frac{(x - 6) (- (x - 6) - 2 (- x - 6))}{(x - 6) {(x - 6)}^{3}} + \frac{d}{d x} [9]$

Step 1.1.2.19.2

Cancel the common factor.

$\frac{(x - 6) (- (x - 6) - 2 (- x - 6))}{(x - 6) {(x - 6)}^{3}} + \frac{d}{d x} [9]$

Step 1.1.2.19.3

Rewrite the expression.

$\frac{- (x - 6) - 2 (- x - 6)}{{(x - 6)}^{3}} + \frac{d}{d x} [9]$

Step 1.1.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Apply the distributive property.

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

Step 1.1.4.3

Combine terms.

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Step 1.1.4.3.1

Multiply $- 1$ by $- 6$ .

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

Step 1.1.4.3.2

Multiply $- 1$ by $- 2$ .

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

Step 1.1.4.3.3

Multiply $- 2$ by $- 6$ .

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

Step 1.1.4.3.4

Add $- x$ and $2 x$ .

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

Step 1.1.4.3.5

Add $6$ and $12$ .

$\frac{x + 18}{{(x - 6)}^{3}} + 0$

Step 1.1.4.3.6

Add $\frac{x + 18}{{(x - 6)}^{3}}$ and $0$ .

$f' (x) = \frac{x + 18}{{(x - 6)}^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x + 18}{{(x - 6)}^{3}}$ .

$\frac{x + 18}{{(x - 6)}^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x + 18}{{(x - 6)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x + 18}{{(x - 6)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$x + 18 = 0$

Step 2.3

Subtract $18$ from both sides of the equation.

$x = - 18$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x + 18}{{(x - 6)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x - 6)}^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Evaluate $\frac{- x - 6}{{(x - 6)}^{2}} + 9$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 18$ .

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Step 4.1.1

Substitute $- 18$ for $x$ .

$\frac{- (- 18) - 6}{{((- 18) - 6)}^{2}} + 9$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the expression.

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Step 4.1.2.1.1

Multiply $- 1$ by $- 18$ .

$\frac{18 - 6}{{((- 18) - 6)}^{2}} + 9$

Step 4.1.2.1.2

Subtract $6$ from $18$ .

$\frac{12}{{((- 18) - 6)}^{2}} + 9$

Step 4.1.2.2

Simplify each term.

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Step 4.1.2.2.1

Simplify the denominator.

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Step 4.1.2.2.1.1

Subtract $6$ from $- 18$ .

$\frac{12}{{(- 24)}^{2}} + 9$

Step 4.1.2.2.1.2

Raise $- 24$ to the power of $2$ .

$\frac{12}{576} + 9$

Step 4.1.2.2.2

Cancel the common factor of $12$ and $576$ .

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Step 4.1.2.2.2.1

Factor $12$ out of $12$ .

$\frac{12 (1)}{576} + 9$

Step 4.1.2.2.2.2

Cancel the common factors.

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Step 4.1.2.2.2.2.1

Factor $12$ out of $576$ .

$\frac{12 \cdot 1}{12 \cdot 48} + 9$

Step 4.1.2.2.2.2.2

Cancel the common factor.

$\frac{12 \cdot 1}{12 \cdot 48} + 9$

Step 4.1.2.2.2.2.3

Rewrite the expression.

$\frac{1}{48} + 9$

Step 4.1.2.3

To write $9$ as a fraction with a common denominator, multiply by $\frac{48}{48}$ .

$\frac{1}{48} + 9 \cdot \frac{48}{48}$

Step 4.1.2.4

Combine $9$ and $\frac{48}{48}$ .

$\frac{1}{48} + \frac{9 \cdot 48}{48}$

Step 4.1.2.5

Combine the numerators over the common denominator.

$\frac{1 + 9 \cdot 48}{48}$

Step 4.1.2.6

Simplify the numerator.

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Step 4.1.2.6.1

Multiply $9$ by $48$ .

$\frac{1 + 432}{48}$

Step 4.1.2.6.2

Add $1$ and $432$ .

$\frac{433}{48}$

Step 4.2

Evaluate at $x = 6$ .

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Step 4.2.1

Substitute $6$ for $x$ .

$\frac{- (6) - 6}{{((6) - 6)}^{2}} + 9$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Subtract $6$ from $6$ .

$\frac{- (6) - 6}{0^{2}} + 9$

Step 4.2.2.2

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

$\frac{- (6) - 6}{0} + 9$

Step 4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(- 18, \frac{433}{48})$

Step 5