Calculus Examples

$2 + | x - 3 |$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $2 + | x - 3 |$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [| x - 3 |]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [| x - 3 |]$

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [| x - 3 |]$

Step 2

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

Tap for more steps...

Step 2.1

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 $g (x) = x - 3$ .

Tap for more steps...

Step 2.1.1

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

$0 + \frac{d}{d u} [| u |] \frac{d}{d x} [x - 3]$

Step 2.1.2

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

$0 + \frac{u}{| u |} \frac{d}{d x} [x - 3]$

Step 2.1.3

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

$0 + \frac{x - 3}{| x - 3 |} \frac{d}{d x} [x - 3]$

Step 2.2

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

$0 + \frac{x - 3}{| x - 3 |} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.3

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

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

Step 2.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 2.5

Add $1$ and $0$ .

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

Step 2.6

Multiply $\frac{x - 3}{| x - 3 |}$ by $1$ .

$0 + \frac{x - 3}{| x - 3 |}$

Step 3

Add $0$ and $\frac{x - 3}{| x - 3 |}$ .

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