Calculus Examples

$f (x) = x^{3} - 12 x + 1$

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} - 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x] + \frac{d}{d x} [1]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (1)$

Step 1.1.1.2

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (- 12 x) + \frac{d}{d x} (1)$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Step 1.1.2.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} - 12 \frac{d}{d x} x + \frac{d}{d x} (1)$

Step 1.1.2.2

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

$f' (x) = 3 x^{2} - 12 \cdot 1 + \frac{d}{d x} (1)$

Step 1.1.2.3

Multiply $- 12$ by $1$ .

$f' (x) = 3 x^{2} - 12 + \frac{d}{d x} (1)$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f' (x) = 3 x^{2} - 12 + 0$

Step 1.1.3.2

Add $3 x^{2} - 12$ and $0$ .

$f' (x) = 3 x^{2} - 12$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 12$ .

$3 x^{2} - 12$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 12 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 12 = 0$

Step 2.2

Add $12$ to both sides of the equation.

$3 x^{2} = 12$

Step 2.3

Divide each term in $3 x^{2} = 12$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 x^{2} = 12$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{12}{3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{12}{3}$

Step 2.3.2.1.2

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

$x^{2} = \frac{12}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $12$ by $3$ .

$x^{2} = 4$

Step 2.4

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

$x = \pm \sqrt{4}$

Step 2.5

Simplify $\pm \sqrt{4}$ .

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Step 2.5.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.5.2

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

$x = \pm 2$

Step 2.6

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

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Step 2.6.1

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

$x = 2$

Step 2.6.2

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

$x = - 2$

Step 2.6.3

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

$x = 2, - 2$

Step 3

The values which make the derivative equal to $0$ are $2, - 2$ .

$2, - 2$

Step 4

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

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

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

$f' (- 3) = 3 {(- 3)}^{2} - 12$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (- 3) = 3 \cdot 9 - 12$

Step 5.2.1.2

Multiply $3$ by $9$ .

$f' (- 3) = 27 - 12$

Step 5.2.2

Subtract $12$ from $27$ .

$f' (- 3) = 15$

Step 5.2.3

The final answer is $15$ .

$15$

Step 5.3

At $x = - 3$ the derivative is $15$ . Since this is positive, the function is increasing on $(- \infty, - 2)$ .

Increasing on $(- \infty, - 2)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 2, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = 3 {(0)}^{2} - 12$

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

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

$f' (0) = 3 \cdot 0 - 12$

Step 6.2.1.2

Multiply $3$ by $0$ .

$f' (0) = 0 - 12$

Step 6.2.2

Subtract $12$ from $0$ .

$f' (0) = - 12$

Step 6.2.3

The final answer is $- 12$ .

$- 12$

Step 6.3

At $x = 0$ the derivative is $- 12$ . Since this is negative, the function is decreasing on $(- 2, 2)$ .

Decreasing on $(- 2, 2)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (3) = 3 {(3)}^{2} - 12$

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

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 7.2.1.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 7.2.1.1.1.1

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

$f' (3) = 3 {(3)}^{2} - 12$

Step 7.2.1.1.1.2

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

$f' (3) = 3^{1 + 2} - 12$

Step 7.2.1.1.2

Add $1$ and $2$ .

$f' (3) = 3^{3} - 12$

Step 7.2.1.2

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

$f' (3) = 27 - 12$

Step 7.2.2

Subtract $12$ from $27$ .

$f' (3) = 15$

Step 7.2.3

The final answer is $15$ .

$15$

Step 7.3

At $x = 3$ the derivative is $15$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, - 2), (2, \infty)$

Decreasing on: $(- 2, 2)$

Step 9