Calculus Examples

$x^{3} - 27 x$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} - 27 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 27 x]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 27 x]$

Step 1.1.2

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

$3 x^{2} + \frac{d}{d x} [- 27 x]$

Step 1.2

Evaluate $\frac{d}{d x} [- 27 x]$ .

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Step 1.2.1

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27 x$ with respect to $x$ is $- 27 \frac{d}{d x} [x]$ .

$3 x^{2} - 27 \frac{d}{d x} [x]$

Step 1.2.2

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

$3 x^{2} - 27 \cdot 1$

Step 1.2.3

Multiply $- 27$ by $1$ .

$3 x^{2} - 27$

Step 2

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 27 = 0$ .

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Step 2.1

Add $27$ to both sides of the equation.

$3 x^{2} = 27$

Step 2.2

Divide each term in $3 x^{2} = 27$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x^{2} = 27$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{27}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{27}{3}$

Step 2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{27}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $27$ by $3$ .

$x^{2} = 9$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Step 2.4

Simplify $\pm \sqrt{9}$ .

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Step 2.4.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 2.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 3

Solve the original function $f (x) = x^{3} - 27 x$ at $x = 3$ .

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Step 3.1

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

$f (3) = {(3)}^{3} - 27 \cdot 3$

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 $3$ .

$f (3) = 27 - 27 \cdot 3$

Step 3.2.1.2

Multiply $- 27$ by $3$ .

$f (3) = 27 - 81$

Step 3.2.2

Subtract $81$ from $27$ .

$f (3) = - 54$

Step 3.2.3

The final answer is $- 54$ .

$- 54$

Step 4

Solve the original function $f (x) = x^{3} - 27 x$ at $x = - 3$ .

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Step 4.1

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

$f (- 3) = {(- 3)}^{3} - 27 \cdot - 3$

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

$f (- 3) = - 27 - 27 \cdot - 3$

Step 4.2.1.2

Multiply $- 27$ by $- 3$ .

$f (- 3) = - 27 + 81$

Step 4.2.2

Add $- 27$ and $81$ .

$f (- 3) = 54$

Step 4.2.3

The final answer is $54$ .

$54$

Step 5

The horizontal tangent lines on function $f (x) = x^{3} - 27 x$ are $y = - 54, y = 54$ .

$y = - 54, y = 54$

Step 6