# How do I use one sided derivatives to show that the function #f(x)=x^3 , x<=1# #f(x)=3x , x>1# is not differentiable at x=1?

##### 1 Answer

Actually, it seems to me that the function **is** differentiable

#### Explanation:

This is a piecewise function, i.e. a function defined with different rules in different intervals.

In this case, both rules

But what does this mean? Well, a function is differentiable if the right and left derivatives exist, and are the same.

So, inside both domains

This means that the only point in which this equation may break is when you pass from one domain to the other, i.e. when you switch from one definition to the other.

So, if

On the other hand, if

So, we've proven that the left and right derivative of

If