You are thinking about differentiation only mechanically, without any thought of what it means. The derivative of a function at a point is the limit of the difference quotient there, and is the slope of the tangent line there. So, draw the graphs of $y=x^r$ for various $r$.
For $r=1$, one has a line, and the tangent is the line itself, with slope $1$.
For $r=2$, one has a parabola, and the tangent at $(0, 0)$ is the $x$-axis, with slope $0$.
For $r=\frac12$, one has what turns out to be half the parabola $x=y^2$, and the tangent line is vertical, with infinite slope. Also, this is not defined for $x < 0$, so the limit does not exist anyway.
For $x=\frac13$, the function exists for all $x$, but the tangent at $(0,0)$ is still vertical, the slope is infinite, and the derivative does not exist at $0$.