Calculus Examples

$f (x) = 5 - | x - 5 |$

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 $5 - | x - 5 |$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- | x - 5 |]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- | x - 5 |]$

Step 1.1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- | x - 5 |]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- | x - 5 |]$ .

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- | x - 5 |$ with respect to $x$ is $- \frac{d}{d x} [| x - 5 |]$ .

$0 - \frac{d}{d x} [| x - 5 |]$

Step 1.1.2.2

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 $g (x) = x - 5$ .

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Step 1.1.2.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$0 - (\frac{d}{d u} [| u |] \frac{d}{d x} [x - 5])$

Step 1.1.2.2.2

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

$0 - (\frac{u}{| u |} \frac{d}{d x} [x - 5])$

Step 1.1.2.2.3

Replace all occurrences of $u$ with $x - 5$ .

$0 - (\frac{x - 5}{| x - 5 |} \frac{d}{d x} [x - 5])$

Step 1.1.2.3

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$0 - (\frac{x - 5}{| x - 5 |} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5]))$

Step 1.1.2.4

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

$0 - (\frac{x - 5}{| x - 5 |} (1 + \frac{d}{d x} [- 5]))$

Step 1.1.2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$0 - (\frac{x - 5}{| x - 5 |} (1 + 0))$

Step 1.1.2.6

Add $1$ and $0$ .

$0 - (\frac{x - 5}{| x - 5 |} \cdot 1)$

Step 1.1.2.7

Multiply $\frac{x - 5}{| x - 5 |}$ by $1$ .

$0 - \frac{x - 5}{| x - 5 |}$

Step 1.1.3

Subtract $\frac{x - 5}{| x - 5 |}$ from $0$ .

$f' (x) = - \frac{x - 5}{| x - 5 |}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x - 5}{| x - 5 |}$ .

$- \frac{x - 5}{| x - 5 |}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x - 5}{| x - 5 |} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x - 5}{| x - 5 |} = 0$

Step 2.2

Set the numerator equal to zero.

$x - 5 = 0$

Step 2.3

Add $5$ to both sides of the equation.

$x = 5$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x - 5}{| x - 5 |}$ equal to $0$ to find where the expression is undefined.

$| x - 5 | = 0$

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 $| x | = \pm x$ .

$x - 5 = \pm 0$

Step 3.2.2

Plus or minus $0$ is $0$ .

$x - 5 = 0$

Step 3.2.3

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Evaluate $5 - | x - 5 |$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 5$ .

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Step 4.1.1

Substitute $5$ for $x$ .

$5 - | (5) - 5 |$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Subtract $5$ from $5$ .

$5 - | 0 |$

Step 4.1.2.1.2

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

$5 - 0$

Step 4.1.2.1.3

Multiply $- 1$ by $0$ .

$5 + 0$

Step 4.1.2.2

Add $5$ and $0$ .

$5$

Step 4.2

List all of the points.

$(5, 5)$

Step 5