Definition of Differentiable
- the function has derivative
- the derivative f’(x) must exist for every value in the function’s domain
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
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)
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)
Integration by Substitution (Reverse Chain Rule)