Calculus Examples

$f (x) = 4 x^{3} - 48 x$

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 $4 x^{3} - 48 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 48 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 48 x]$

Step 1.1.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 48 x]$

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 $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [- 48 x]$

Step 1.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 48 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 48 x]$ .

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Step 1.1.3.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$12 x^{2} - 48 \frac{d}{d x} [x]$

Step 1.1.3.2

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

$12 x^{2} - 48 \cdot 1$

Step 1.1.3.3

Multiply $- 48$ by $1$ .

$f' (x) = 12 x^{2} - 48$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 48$ .

$12 x^{2} - 48$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 48 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} - 48 = 0$

Step 2.2

Add $48$ to both sides of the equation.

$12 x^{2} = 48$

Step 2.3

Divide each term in $12 x^{2} = 48$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 x^{2} = 48$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{48}{12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{48}{12}$

Step 2.3.2.1.2

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

$x^{2} = \frac{48}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $48$ by $12$ .

$x^{2} = 4$

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.

$f' (- 3) = 12 {(- 3)}^{2} - 48$

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 $2$ .

$f' (- 3) = 12 \cdot 9 - 48$

Step 5.2.1.2

Multiply $12$ by $9$ .

$f' (- 3) = 108 - 48$

Step 5.2.2

Subtract $48$ from $108$ .

$f' (- 3) = 60$

Step 5.2.3

The final answer is $60$ .

$60$

Step 5.3

At $x = - 3$ the derivative is $60$ . 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.

$f' (0) = 12 {(0)}^{2} - 48$

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 $0$ .

$f' (0) = 12 \cdot 0 - 48$

Step 6.2.1.2

Multiply $12$ by $0$ .

$f' (0) = 0 - 48$

Step 6.2.2

Subtract $48$ from $0$ .

$f' (0) = - 48$

Step 6.2.3

The final answer is $- 48$ .

$- 48$

Step 6.3

At $x = 0$ the derivative is $- 48$ . 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.

$f' (3) = 12 {(3)}^{2} - 48$

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 $3$ to the power of $2$ .

$f' (3) = 12 \cdot 9 - 48$

Step 7.2.1.2

Multiply $12$ by $9$ .

$f' (3) = 108 - 48$

Step 7.2.2

Subtract $48$ from $108$ .

$f' (3) = 60$

Step 7.2.3

The final answer is $60$ .

$60$

Step 7.3

At $x = 3$ the derivative is $60$ . 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