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It is shown that, under certain assumptions on the growth and valuedistribution of a meromorphic function $f(z)$, \begin{equation*}m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where$\Delta_c f=f(z+c)-f(z)$ and $a,c\in\mathbb{C}$. This estimate implies a lowerbound for the Nevanlinna ramification term in terms of the difference operatorwith an arbitrary shift. As a consequence it follows, for instance, that if $f$is an entire function of hyper-order $<1$ whose derivative does not attain avalue $a\in\mathbb{C}$ often $$N\left(r,\frac{1}{f'-a}\right)=S(r,f),$$ thenthe finite difference $\Delta_c f$ cannot attain the value $ac$ significantlymore often $$N\left(r,\frac{1}{\Delta_c f-ac}\right)=S(r,f).$$ Additionalapplications of the estimate above include a new type of a second main theorem,deficiency relations between $\Delta_cf$ and $f'$ and new Clunie and Mohon'kotype lemmas.

A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function