Calculus Examples

$f (x) = | 3 x - 4 |$

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 $g (x) = 3 x - 4$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $3 x - 4$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [3 x - 4]$

Step 1.1.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [3 x - 4]$

Step 1.1.1.3

Replace all occurrences of $u$ with $3 x - 4$ .

$\frac{3 x - 4}{| 3 x - 4 |} \frac{d}{d x} [3 x - 4]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $3 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

$\frac{3 x - 4}{| 3 x - 4 |} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4])$

Step 1.1.2.2

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

$\frac{3 x - 4}{| 3 x - 4 |} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4])$

Step 1.1.2.3

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

$\frac{3 x - 4}{| 3 x - 4 |} (3 \cdot 1 + \frac{d}{d x} [- 4])$

Step 1.1.2.4

Multiply $3$ by $1$ .

$\frac{3 x - 4}{| 3 x - 4 |} (3 + \frac{d}{d x} [- 4])$

Step 1.1.2.5

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

$\frac{3 x - 4}{| 3 x - 4 |} (3 + 0)$

Step 1.1.2.6

Combine fractions.

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Step 1.1.2.6.1

Add $3$ and $0$ .

$\frac{3 x - 4}{| 3 x - 4 |} \cdot 3$

Step 1.1.2.6.2

Combine $\frac{3 x - 4}{| 3 x - 4 |}$ and $3$ .

$\frac{(3 x - 4) \cdot 3}{| 3 x - 4 |}$

Step 1.1.2.6.3

Move $3$ to the left of $3 x - 4$ .

$\frac{3 \cdot (3 x - 4)}{| 3 x - 4 |}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$\frac{3 (3 x) + 3 \cdot - 4}{| 3 x - 4 |}$

Step 1.1.3.2

Simplify each term.

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Step 1.1.3.2.1

Multiply $3$ by $3$ .

$\frac{9 x + 3 \cdot - 4}{| 3 x - 4 |}$

Step 1.1.3.2.2

Multiply $3$ by $- 4$ .

$\frac{9 x - 12}{| 3 x - 4 |}$

Step 1.1.3.3

Factor $3$ out of $9 x - 12$ .

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Step 1.1.3.3.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 12}{| 3 x - 4 |}$

Step 1.1.3.3.2

Factor $3$ out of $- 12$ .

$\frac{3 (3 x) + 3 (- 4)}{| 3 x - 4 |}$

Step 1.1.3.3.3

Factor $3$ out of $3 (3 x) + 3 (- 4)$ .

$f' (x) = \frac{3 (3 x - 4)}{| 3 x - 4 |}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 (3 x - 4)}{| 3 x - 4 |}$ .

$\frac{3 (3 x - 4)}{| 3 x - 4 |}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{3 (3 x - 4)}{| 3 x - 4 |} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{3 (3 x - 4)}{| 3 x - 4 |} = 0$

Step 2.2

Set the numerator equal to zero.

$3 (3 x - 4) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $3 (3 x - 4) = 0$ by $3$ and simplify.

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Step 2.3.1.1

Divide each term in $3 (3 x - 4) = 0$ by $3$ .

$\frac{3 (3 x - 4)}{3} = \frac{0}{3}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $3$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{3 (3 x - 4)}{3} = \frac{0}{3}$

Step 2.3.1.2.1.2

Divide $3 x - 4$ by $1$ .

$3 x - 4 = \frac{0}{3}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $3$ .

$3 x - 4 = 0$

Step 2.3.2

Add $4$ to both sides of the equation.

$3 x = 4$

Step 2.3.3

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 2.3.3.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{3 (3 x - 4)}{| 3 x - 4 |}$ equal to $0$ to find where the expression is undefined.

$| 3 x - 4 | = 0$

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 $| x | = \pm x$ .

$3 x - 4 = \pm 0$

Step 3.2.2

Plus or minus $0$ is $0$ .

$3 x - 4 = 0$

Step 3.2.3

Add $4$ to both sides of the equation.

$3 x = 4$

Step 3.2.4

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 3.2.4.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $3$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 4

Evaluate $| 3 x - 4 |$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{4}{3}$ .

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Step 4.1.1

Substitute $\frac{4}{3}$ for $x$ .

$| 3 (\frac{4}{3}) - 4 |$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.1.1

Cancel the common factor.

$| 3 (\frac{4}{3}) - 4 |$

Step 4.1.2.1.2

Rewrite the expression.

$| 4 - 4 |$

Step 4.1.2.2

Subtract $4$ from $4$ .

$| 0 |$

Step 4.1.2.3

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

$0$

Step 4.2

List all of the points.

$(\frac{4}{3}, 0)$

Step 5