Assume that $ f$ is differentiable on $ ]x,x+\delta[$ for some $ \delta>0$ . Consider the following two statements:

(1) $ f$ has a right derivative at $ x$ , that is $ $ f_{+}^{\prime}(x)=\lim_{\epsilon\rightarrow 0^{+}}\frac{f(x+\epsilon)-f(x)}{\epsilon}$ $ exists.

(2) $ \lim_{y\rightarrow x^{+}}f^{\prime}(y)$ exists.

Which statement is stronger. In other words,

does (1) imply (2) but (2) does not imply (1),

or does (2) imply (1) but (1) does not imply (2),

or are they equivalent ?