Calculus Examples

$f (x) = 4 - | x |$

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $4 - | x |$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- | x |]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- | x |]$

Step 1.1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x = 0$

Step 2.3

Exclude the solutions that do not make $- \frac{x}{| x |} = 0$ true.

$No solution$

Step 3

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 4

Find where the derivative is undefined.

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Step 4.1

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

$| x | = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

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

$x = \pm 0$

Step 4.2.2

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

After finding the point that makes the derivative $f' (x) = - \frac{x}{| x |}$ equal to $0$ or undefined, the interval to check where $f (x) = 4 - | x |$ is increasing and where it is decreasing is $(- \infty, 0) \cup (0, \infty)$ .

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{- 1}{| - 1 |}$

Step 6.2

Simplify the result.

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Step 6.2.1

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$f' (- 1) = - \frac{- 1}{1}$

Step 6.2.2

Divide $- 1$ by $1$ .

$f' (- 1) = 1$

Step 6.2.3

The final answer is $1$ .

$1$

Step 6.3

At $x = - 1$ the derivative is $1$ . Since this is positive, the function is increasing on $(- \infty, 0)$ .

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

Step 7

Substitute a value from the interval $(0, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{1}{| 1 |}$

Step 7.2

Simplify the result.

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Step 7.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$f' (1) = - \frac{1}{1}$

Step 7.2.2

Cancel the common factor of $1$ .

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Step 7.2.2.1

Cancel the common factor.

$f' (1) = - \frac{1}{1}$

Step 7.2.2.2

Rewrite the expression.

$f' (1) = - 1 \cdot 1$

Step 7.2.3

Multiply $- 1$ by $1$ .

$f' (1) = - 1$

Step 7.2.4

The final answer is $- 1$ .

$- 1$

Step 7.3

At $x = 1$ the derivative is $- 1$ . Since this is negative, the function is decreasing on $(0, \infty)$ .

Decreasing on $(0, \infty)$ since $f' (x) < 0$

Step 8

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

Increasing on: $(- \infty, 0)$

Decreasing on: $(0, \infty)$

Step 9