We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $f(u)=0$ and $f(a)\neq 0$ to find a $c\in(u,a)$ such that: the expression $f(a)/(a-u)$ is the slope of the chord of the graph of $f$, and $f'(c)$ is the slope of the tangent line to the curve at the point $(c,f'(c))$.
My question is: How I can define an equivalence relation classifying the different slopes: the slope of the tangent line to the curve at the point $\left(c,f'(c)\right)\,$? I guess that two points $(c,f'(c))$ and $(d,f'(d))$ are equivalent if and only if the slopes of the tangent lines to the curve at the two points are equal. However, I am not able for formulate this relation mathematically.
We define relation $R$ on $\mathbb{R}\times\mathbb{R}$ via $(x_1,y_1)R(x_2,y_2)$ if and only if $y_1=y_2$.