Equivalence of Definitions of Derivative

Theorem
The two forms of the definition of a derivative of a real function at any point $\left({c, f\left({c}\right)}\right)$ are consistent.

That is, for any constant $c$ in the domain of $f$ for which $f\,^\prime \left({c}\right)$ exists:


 * $\displaystyle f\,^\prime \left({c}\right) = \lim_{\Delta x \to 0} \frac {f \left({c + {\Delta x}}\right) - f \left({c}\right)} {\Delta x}$

and:


 * $\displaystyle f\,^\prime \left({c}\right) = \lim_{x \to c} \frac {f \left({x}\right) - f \left({c}\right)} {x - c}$

are logically equivalent.