Calculus Examples

$(x - 9) (x^{2} - 18 x - 162)$

Step 1

Write $(x - 9) (x^{2} - 18 x - 162)$ as a function.

$f (x) = (x - 9) (x^{2} - 18 x - 162)$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 9$ and $g (x) = x^{2} - 18 x - 162$ .

$f' (x) = (x - 9) \frac{d}{d x} (x^{2} - 18 x - 162) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $x^{2} - 18 x - 162$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [- 162]$ .

$f' (x) = (x - 9) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 18 x) + \frac{d}{d x} (- 162)) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.2

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

$f' (x) = (x - 9) (2 x + \frac{d}{d x} (- 18 x) + \frac{d}{d x} (- 162)) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.3

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$f' (x) = (x - 9) (2 x - 18 \frac{d}{d x} x + \frac{d}{d x} (- 162)) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.4

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

$f' (x) = (x - 9) (2 x - 18 \cdot 1 + \frac{d}{d x} (- 162)) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.5

Multiply $- 18$ by $1$ .

$f' (x) = (x - 9) (2 x - 18 + \frac{d}{d x} (- 162)) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.6

Since $- 162$ is constant with respect to $x$ , the derivative of $- 162$ with respect to $x$ is $0$ .

$f' (x) = (x - 9) (2 x - 18 + 0) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.7

Add $2 x - 18$ and $0$ .

$f' (x) = (x - 9) (2 x - 18) + (x^{2} - 18 x - 162) \frac{d}{d x} (x - 9)$

Step 2.1.2.8

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

$f' (x) = (x - 9) (2 x - 18) + (x^{2} - 18 x - 162) (\frac{d}{d x} (x) + \frac{d}{d x} (- 9))$

Step 2.1.2.9

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

$f' (x) = (x - 9) (2 x - 18) + (x^{2} - 18 x - 162) (1 + \frac{d}{d x} (- 9))$

Step 2.1.2.10

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

$f' (x) = (x - 9) (2 x - 18) + (x^{2} - 18 x - 162) (1 + 0)$

Step 2.1.2.11

Simplify the expression.

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Step 2.1.2.11.1

Add $1$ and $0$ .

$f' (x) = (x - 9) (2 x - 18) + (x^{2} - 18 x - 162) \cdot 1$

Step 2.1.2.11.2

Multiply $x^{2} - 18 x - 162$ by $1$ .

$f' (x) = (x - 9) (2 x - 18) + x^{2} - 18 x - 162$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Apply the distributive property.

$f' (x) = (x - 9) (2 x) + (x - 9) \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.2

Apply the distributive property.

$f' (x) = x (2 x) - 9 (2 x) + (x - 9) \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.3

Apply the distributive property.

$f' (x) = x (2 x) - 9 (2 x) + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4

Combine terms.

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Step 2.1.3.4.1

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

$f' (x) = 2 (x \cdot x) - 9 (2 x) + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.2

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

$f' (x) = 2 (x \cdot x) - 9 (2 x) + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.3

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

$f' (x) = 2 x^{1 + 1} - 9 (2 x) + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.4

Add $1$ and $1$ .

$f' (x) = 2 x^{2} - 9 (2 x) + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.5

Multiply $2$ by $- 9$ .

$f' (x) = 2 x^{2} - 18 x + x \cdot - 18 - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.6

Move $- 18$ to the left of $x$ .

$f' (x) = 2 x^{2} - 18 x - 18 \cdot x - 9 \cdot - 18 + x^{2} - 18 x - 162$

Step 2.1.3.4.7

Multiply $- 9$ by $- 18$ .

$f' (x) = 2 x^{2} - 18 x - 18 x + 162 + x^{2} - 18 x - 162$

Step 2.1.3.4.8

Subtract $18 x$ from $- 18 x$ .

$f' (x) = 2 x^{2} - 36 x + 162 + x^{2} - 18 x - 162$

Step 2.1.3.4.9

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

$f' (x) = 3 x^{2} - 36 x + 162 - 18 x - 162$

Step 2.1.3.4.10

Subtract $18 x$ from $- 36 x$ .

$f' (x) = 3 x^{2} - 54 x + 162 - 162$

Step 2.1.3.4.11

Subtract $162$ from $162$ .

$f' (x) = 3 x^{2} - 54 x + 0$

Step 2.1.3.4.12

Add $3 x^{2} - 54 x$ and $0$ .

$f' (x) = 3 x^{2} - 54 x$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 54 x$ .

$3 x^{2} - 54 x$

Step 3

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 54 x = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$3 x^{2} - 54 x = 0$

Step 3.2

Factor $3 x$ out of $3 x^{2} - 54 x$ .

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Step 3.2.1

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

$3 x (x) - 54 x = 0$

Step 3.2.2

Factor $3 x$ out of $- 54 x$ .

$3 x (x) + 3 x (- 18) = 0$

Step 3.2.3

Factor $3 x$ out of $3 x (x) + 3 x (- 18)$ .

$3 x (x - 18) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 18 = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $x - 18$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Step 3.5.2

Add $18$ to both sides of the equation.

$x = 18$

Step 3.6

The final solution is all the values that make $3 x (x - 18) = 0$ true.

$x = 0, 18$

Step 4

The values which make the derivative equal to $0$ are $0, 18$ .

$0, 18$

Step 5

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 18) \cup (18, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 3 {(- 1)}^{2} - 54 \cdot - 1$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Raise $- 1$ to the power of $2$ .

$f' (- 1) = 3 \cdot 1 - 54 \cdot - 1$

Step 6.2.1.2

Multiply $3$ by $1$ .

$f' (- 1) = 3 - 54 \cdot - 1$

Step 6.2.1.3

Multiply $- 54$ by $- 1$ .

$f' (- 1) = 3 + 54$

Step 6.2.2

Add $3$ and $54$ .

$f' (- 1) = 57$

Step 6.2.3

The final answer is $57$ .

$57$

Step 6.3

At $x = - 1$ the derivative is $57$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f' (x) > 0$

Step 7

Substitute a value from the interval $(0, 18)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = 3 {(9)}^{2} - 54 \cdot 9$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $9$ to the power of $2$ .

$f' (9) = 3 \cdot 81 - 54 \cdot 9$

Step 7.2.1.2

Multiply $3$ by $81$ .

$f' (9) = 243 - 54 \cdot 9$

Step 7.2.1.3

Multiply $- 54$ by $9$ .

$f' (9) = 243 - 486$

Step 7.2.2

Subtract $486$ from $243$ .

$f' (9) = - 243$

Step 7.2.3

The final answer is $- 243$ .

$- 243$

Step 7.3

At $x = 9$ the derivative is $- 243$ . Since this is negative, the function is decreasing on $(0, 18)$ .

Decreasing on $(0, 18)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(18, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $19$ in the expression.

$f' (19) = 3 {(19)}^{2} - 54 \cdot 19$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $19$ to the power of $2$ .

$f' (19) = 3 \cdot 361 - 54 \cdot 19$

Step 8.2.1.2

Multiply $3$ by $361$ .

$f' (19) = 1083 - 54 \cdot 19$

Step 8.2.1.3

Multiply $- 54$ by $19$ .

$f' (19) = 1083 - 1026$

Step 8.2.2

Subtract $1026$ from $1083$ .

$f' (19) = 57$

Step 8.2.3

The final answer is $57$ .

$57$

Step 8.3

At $x = 19$ the derivative is $57$ . Since this is positive, the function is increasing on $(18, \infty)$ .

Increasing on $(18, \infty)$ since $f' (x) > 0$

Step 9

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, 0), (18, \infty)$

Decreasing on: $(0, 18)$

Step 10