Calculus Examples

$| x | - 3 | x + 1 |$

Step 1

By the Sum Rule, the derivative of $| x | - 3 | x + 1 |$ with respect to $x$ is $\frac{d}{d x} [| x |] + \frac{d}{d x} [- 3 | x + 1 |]$ .

$\frac{d}{d x} [| x |] + \frac{d}{d x} [- 3 | x + 1 |]$

Step 2

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

$\frac{x}{| x |} + \frac{d}{d x} [- 3 | x + 1 |]$

Step 3

Evaluate $\frac{d}{d x} [- 3 | x + 1 |]$ .

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 | x + 1 |$ with respect to $x$ is $- 3 \frac{d}{d x} [| x + 1 |]$ .

$\frac{x}{| x |} - 3 \frac{d}{d x} [| x + 1 |]$

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 $g (x) = x + 1$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

$\frac{x}{| x |} - 3 (\frac{x + 1}{| x + 1 |} (1 + 0))$

Step 3.6

Add $1$ and $0$ .

$\frac{x}{| x |} - 3 (\frac{x + 1}{| x + 1 |} \cdot 1)$

Step 3.7

Multiply $\frac{x + 1}{| x + 1 |}$ by $1$ .

$\frac{x}{| x |} - 3 \frac{x + 1}{| x + 1 |}$

Step 3.8

Combine $- 3$ and $\frac{x + 1}{| x + 1 |}$ .

$\frac{x}{| x |} + \frac{- 3 (x + 1)}{| x + 1 |}$

Step 3.9

Move the negative in front of the fraction.

$\frac{x}{| x |} - \frac{3 (x + 1)}{| x + 1 |}$