Calculus Examples

$6 x^{3} - 18 x^{2} - 54 x + 18$

Step 1

Find the derivative.

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Step 1.1

By the Sum Rule, the derivative of $6 x^{3} - 18 x^{2} - 54 x + 18$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$ .

$\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.2

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$6 (3 x^{2}) + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.2.3

Multiply $3$ by $6$ .

$18 x^{2} + \frac{d}{d x} [- 18 x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.3

Evaluate $\frac{d}{d x} [- 18 x^{2}]$ .

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Step 1.3.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x^{2}$ with respect to $x$ is $- 18 \frac{d}{d x} [x^{2}]$ .

$18 x^{2} - 18 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.3.2

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

$18 x^{2} - 18 (2 x) + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.3.3

Multiply $2$ by $- 18$ .

$18 x^{2} - 36 x + \frac{d}{d x} [- 54 x] + \frac{d}{d x} [18]$

Step 1.4

Evaluate $\frac{d}{d x} [- 54 x]$ .

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Step 1.4.1

Since $- 54$ is constant with respect to $x$ , the derivative of $- 54 x$ with respect to $x$ is $- 54 \frac{d}{d x} [x]$ .

$18 x^{2} - 36 x - 54 \frac{d}{d x} [x] + \frac{d}{d x} [18]$

Step 1.4.2

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

$18 x^{2} - 36 x - 54 \cdot 1 + \frac{d}{d x} [18]$

Step 1.4.3

Multiply $- 54$ by $1$ .

$18 x^{2} - 36 x - 54 + \frac{d}{d x} [18]$

Step 1.5

Differentiate using the Constant Rule.

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Step 1.5.1

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$18 x^{2} - 36 x - 54 + 0$

Step 1.5.2

Add $18 x^{2} - 36 x - 54$ and $0$ .

$18 x^{2} - 36 x - 54$

Step 2

Set the derivative equal to $0$ then solve the equation $18 x^{2} - 36 x - 54 = 0$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $18$ out of $18 x^{2} - 36 x - 54$ .

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Step 2.1.1.1

Factor $18$ out of $18 x^{2}$ .

$18 (x^{2}) - 36 x - 54 = 0$

Step 2.1.1.2

Factor $18$ out of $- 36 x$ .

$18 (x^{2}) + 18 (- 2 x) - 54 = 0$

Step 2.1.1.3

Factor $18$ out of $- 54$ .

$18 x^{2} + 18 (- 2 x) + 18 \cdot - 3 = 0$

Step 2.1.1.4

Factor $18$ out of $18 x^{2} + 18 (- 2 x)$ .

$18 (x^{2} - 2 x) + 18 \cdot - 3 = 0$

Step 2.1.1.5

Factor $18$ out of $18 (x^{2} - 2 x) + 18 \cdot - 3$ .

$18 (x^{2} - 2 x - 3) = 0$

Step 2.1.2

Factor.

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Step 2.1.2.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.1.2.1.2

Write the factored form using these integers.

$18 ((x - 3) (x + 1)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$18 (x - 3) (x + 1) = 0$

Step 2.2

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

$x - 3 = 0$

$x + 1 = 0$

Step 2.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.5

The final solution is all the values that make $18 (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 3

Solve the original function $f (x) = 6 x^{3} - 18 x^{2} - 54 x + 18$ at $x = 3$ .

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 6 {(3)}^{3} - 18 {(3)}^{2} - 54 \cdot 3 + 18$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $3$ to the power of $3$ .

$f (3) = 6 \cdot 27 - 18 {(3)}^{2} - 54 \cdot 3 + 18$

Step 3.2.1.2

Multiply $6$ by $27$ .

$f (3) = 162 - 18 {(3)}^{2} - 54 \cdot 3 + 18$

Step 3.2.1.3

Raise $3$ to the power of $2$ .

$f (3) = 162 - 18 \cdot 9 - 54 \cdot 3 + 18$

Step 3.2.1.4

Multiply $- 18$ by $9$ .

$f (3) = 162 - 162 - 54 \cdot 3 + 18$

Step 3.2.1.5

Multiply $- 54$ by $3$ .

$f (3) = 162 - 162 - 162 + 18$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $162$ from $162$ .

$f (3) = 0 - 162 + 18$

Step 3.2.2.2

Subtract $162$ from $0$ .

$f (3) = - 162 + 18$

Step 3.2.2.3

Add $- 162$ and $18$ .

$f (3) = - 144$

Step 3.2.3

The final answer is $- 144$ .

$- 144$

Step 4

Solve the original function $f (x) = 6 x^{3} - 18 x^{2} - 54 x + 18$ at $x = - 1$ .

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Step 4.1

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

$f (- 1) = 6 {(- 1)}^{3} - 18 {(- 1)}^{2} - 54 \cdot - 1 + 18$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 1$ to the power of $3$ .

$f (- 1) = 6 \cdot - 1 - 18 {(- 1)}^{2} - 54 \cdot - 1 + 18$

Step 4.2.1.2

Multiply $6$ by $- 1$ .

$f (- 1) = - 6 - 18 {(- 1)}^{2} - 54 \cdot - 1 + 18$

Step 4.2.1.3

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

$f (- 1) = - 6 - 18 \cdot 1 - 54 \cdot - 1 + 18$

Step 4.2.1.4

Multiply $- 18$ by $1$ .

$f (- 1) = - 6 - 18 - 54 \cdot - 1 + 18$

Step 4.2.1.5

Multiply $- 54$ by $- 1$ .

$f (- 1) = - 6 - 18 + 54 + 18$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $18$ from $- 6$ .

$f (- 1) = - 24 + 54 + 18$

Step 4.2.2.2

Add $- 24$ and $54$ .

$f (- 1) = 30 + 18$

Step 4.2.2.3

Add $30$ and $18$ .

$f (- 1) = 48$

Step 4.2.3

The final answer is $48$ .

$48$

Step 5

The horizontal tangent lines on function $f (x) = 6 x^{3} - 18 x^{2} - 54 x + 18$ are $y = - 144, y = 48$ .

$y = - 144, y = 48$

Step 6