Calculus Examples

$f (x) = | x^{2} - 16 |$

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 using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 16$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 16$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 16]$

Step 1.1.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - 16]$

Step 1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 16$ .

$\frac{x^{2} - 16}{| x^{2} - 16 |} \frac{d}{d x} [x^{2} - 16]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

$\frac{x^{2} - 16}{| x^{2} - 16 |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16])$

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 $n = 2$ .

$\frac{x^{2} - 16}{| x^{2} - 16 |} (2 x + \frac{d}{d x} [- 16])$

Step 1.1.2.3

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{x^{2} - 16}{| x^{2} - 16 |} (2 x + 0)$

Step 1.1.2.4

Combine fractions.

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Step 1.1.2.4.1

Add $2 x$ and $0$ .

$\frac{x^{2} - 16}{| x^{2} - 16 |} (2 x)$

Step 1.1.2.4.2

Combine $2$ and $\frac{x^{2} - 16}{| x^{2} - 16 |}$ .

$\frac{2 (x^{2} - 16)}{| x^{2} - 16 |} x$

Step 1.1.2.4.3

Combine $\frac{2 (x^{2} - 16)}{| x^{2} - 16 |}$ and $x$ .

$\frac{2 (x^{2} - 16) x}{| x^{2} - 16 |}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot - 16) x}{| x^{2} - 16 |}$

Step 1.1.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot - 16 x}{| x^{2} - 16 |}$

Step 1.1.3.3

Simplify each term.

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Step 1.1.3.3.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot - 16 x}{| x^{2} - 16 |}$

Step 1.1.3.3.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.3.3.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot - 16 x}{| x^{2} - 16 |}$

Step 1.1.3.3.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot - 16 x}{| x^{2} - 16 |}$

Step 1.1.3.3.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot - 16 x}{| x^{2} - 16 |}$

Step 1.1.3.3.2

Multiply $2$ by $- 16$ .

$f' (x) = \frac{2 x^{3} - 32 x}{| x^{2} - 16 |}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x^{3} - 32 x}{| x^{2} - 16 |}$ .

$\frac{2 x^{3} - 32 x}{| x^{2} - 16 |}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2 x^{3} - 32 x}{| x^{2} - 16 |} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2 x^{3} - 32 x}{| x^{2} - 16 |} = 0$

Step 2.2

Set the numerator equal to zero.

$2 x^{3} - 32 x = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Factor the left side of the equation.

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Step 2.3.1.1

Factor $2 x$ out of $2 x^{3} - 32 x$ .

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Step 2.3.1.1.1

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

$2 x (x^{2}) - 32 x = 0$

Step 2.3.1.1.2

Factor $2 x$ out of $- 32 x$ .

$2 x (x^{2}) + 2 x (- 16) = 0$

Step 2.3.1.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 16)$ .

$2 x (x^{2} - 16) = 0$

Step 2.3.1.2

Rewrite $16$ as $4^{2}$ .

$2 x (x^{2} - 4^{2}) = 0$

Step 2.3.1.3

Factor.

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Step 2.3.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$2 x ((x + 4) (x - 4)) = 0$

Step 2.3.1.3.2

Remove unnecessary parentheses.

$2 x (x + 4) (x - 4) = 0$

Step 2.3.2

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

$x = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 2.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 2.3.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.3.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.3.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.3.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.3.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.3.6

The final solution is all the values that make $2 x (x + 4) (x - 4) = 0$ true.

$x = 0, - 4, 4$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{2 x^{3} - 32 x}{| x^{2} - 16 |}$ equal to $0$ to find where the expression is undefined.

$| x^{2} - 16 | = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x^{2} - 16 = \pm 0$

Step 3.2.2

Plus or minus $0$ is $0$ .

$x^{2} - 16 = 0$

Step 3.2.3

Add $16$ to both sides of the equation.

$x^{2} = 16$

Step 3.2.4

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

$x = \pm \sqrt{16}$

Step 3.2.5

Simplify $\pm \sqrt{16}$ .

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Step 3.2.5.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 3.2.5.2

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

$x = \pm 4$

Step 3.2.6

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

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Step 3.2.6.1

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

$x = 4$

Step 3.2.6.2

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

$x = - 4$

Step 3.2.6.3

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

$x = 4, - 4$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 4, x = 4$

Step 4

Evaluate $| x^{2} - 16 |$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$| {(0)}^{2} - 16 |$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$| 0 - 16 |$

Step 4.1.2.2

Subtract $16$ from $0$ .

$| - 16 |$

Step 4.1.2.3

The absolute value is the distance between a number and zero. The distance between $- 16$ and $0$ is $16$ .

$16$

Step 4.2

Evaluate at $x = - 4$ .

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Step 4.2.1

Substitute $- 4$ for $x$ .

$| {(- 4)}^{2} - 16 |$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $- 4$ to the power of $2$ .

$| 16 - 16 |$

Step 4.2.2.2

Subtract $16$ from $16$ .

$| 0 |$

Step 4.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

Step 4.3

Evaluate at $x = 4$ .

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Step 4.3.1

Substitute $4$ for $x$ .

$| {(4)}^{2} - 16 |$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Raise $4$ to the power of $2$ .

$| 16 - 16 |$

Step 4.3.2.2

Subtract $16$ from $16$ .

$| 0 |$

Step 4.3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$0$

Step 4.4

List all of the points.

$(0, 16), (- 4, 0), (4, 0)$

Step 5