## Differentiable

Table of Contents

## Definition of Differentiable

Differentiable means:

- the function has derivative
- the derivative f’(x) must exist for every value in the function’s domain

## Examples

Example 1

1. is x^{3} + 2x differentiable?

- Derivative of x
^{3}+ 2x is**3x**^{2}**+ 2**(the function has derivative) - every value of x has is exist

So x^{3} + 2x is differentiable

Example 2

2. is x^{1/3 }differentiable?

- Derivative of x
^{1/3}is (1/3)x^{-2/3}(the function has derivative) - f’(0) = (1/3)0
^{-2/3 }= undefined

So x^{1/3 }is not differentiable (because the derivative f’(x) doesn’t exist for x=0)

Example 3

3. is x^{-2 }differentiable?

- f(0) = undefined, means f’(0) = undefined

So x^{-2 }is not differentiable (because the derivative f’(x) doesn’t exist for x=0)

The use of differentiable function

When a function is differentiable, we can use all the power of calculus when working with it.

When a function is differentiable, it is continuous.

- Differentiable ⇒ Continuous

- However, a function can be continuous but not differentiable. (example 2)