Calculus Examples

$| x | + 2 | 1 - x |$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 | 1 - x |$ with respect to $x$ is $2 \frac{d}{d x} [| 1 - x |]$ .

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

Step 3.2

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 $g (x) = 1 - x$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Multiply $- 1$ by $1$ .

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

Step 3.8

Subtract $1$ from $0$ .

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Rewrite $- 1 (1 - x)$ as $- (1 - x)$ .

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

Step 3.12

Move the negative in front of the fraction.

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

Step 3.13

Multiply $- 1$ by $2$ .

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

Step 3.14

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

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

Step 3.15

Move the negative in front of the fraction.

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