Calculus Examples

$| 2 x - 1 |$

Step 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 $g (x) = 2 x - 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $2 x - 1$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [2 x - 1]$

Step 1.2

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

$\frac{u}{| u |} \frac{d}{d x} [2 x - 1]$

Step 1.3

Replace all occurrences of $u$ with $2 x - 1$ .

$\frac{2 x - 1}{| 2 x - 1 |} \frac{d}{d x} [2 x - 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{2 x - 1}{| 2 x - 1 |} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])$

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{2 x - 1}{| 2 x - 1 |} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 2.3

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

$\frac{2 x - 1}{| 2 x - 1 |} (2 \cdot 1 + \frac{d}{d x} [- 1])$

Step 2.4

Multiply $2$ by $1$ .

$\frac{2 x - 1}{| 2 x - 1 |} (2 + \frac{d}{d x} [- 1])$

Step 2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{2 x - 1}{| 2 x - 1 |} (2 + 0)$

Step 2.6

Combine fractions.

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Step 2.6.1

Add $2$ and $0$ .

$\frac{2 x - 1}{| 2 x - 1 |} \cdot 2$

Step 2.6.2

Combine $\frac{2 x - 1}{| 2 x - 1 |}$ and $2$ .

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

Step 2.6.3

Move $2$ to the left of $2 x - 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $2$ by $2$ .

$\frac{4 x + 2 \cdot - 1}{| 2 x - 1 |}$

Step 3.2.2

Multiply $2$ by $- 1$ .

$\frac{4 x - 2}{| 2 x - 1 |}$

Step 3.3

Factor $2$ out of $4 x - 2$ .

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Step 3.3.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) - 2}{| 2 x - 1 |}$

Step 3.3.2

Factor $2$ out of $- 2$ .

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

Step 3.3.3

Factor $2$ out of $2 (2 x) + 2 (- 1)$ .

$\frac{2 (2 x - 1)}{| 2 x - 1 |}$