We know that if the tangent line exists then the derivative exists except for vertical tangent lines. For example $f(x)=x^\frac{1}{3}$ has no derivative at $x=0$.
How can we substantiate it both algebracially and intuitively? How about the rate of change or "speed" of the function at this point? The function is continuous and we visually we see that it is growing,but still does it mean the rate of change at $x=0$ is not defined?