## 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 x3 + 2x differentiable?

• Derivative of x3 + 2x is 3x2 + 2 (the function has derivative)
• every value of x has is exist

So x3 + 2x is differentiable

Example 2

2. is x1/3 differentiable?

• Derivative of x1/3 is (1/3)x-2/3 (the function has derivative)
• f’(0) = (1/3)0-2/3 = undefined

So x1/3 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)