# right derivative and right limit of the derivative, which is stronger?

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 ?