Equivalence of Definitions of Derivative

Theorem

The following definitions of the concept of Derivative of Real Function at Point are equivalent:

Let $I$ be an open real interval.

Let $f: I \to \R$ be a real function defined on $I$.

Let $\xi \in I$ be a point in $I$.

Definition 1

That is, suppose the limit $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ exists.

Then this limit is called the derivative of $f$ at the point $\xi$.

Definition 2

That is, suppose the limit $\displaystyle \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$ exists.

Then this limit is called the derivative of $f$ at the point $\xi$.

Proof

 $\displaystyle f' \left({\xi}\right)$ $=$ $\displaystyle \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{x - \xi \mathop \to 0} \frac {f \left({x}\right) - f \left({\xi}\right)} {\xi + h - \xi}$ $\quad$ substituting $x = \xi + h$ $\quad$ $\displaystyle$ $=$ $\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$ $\quad$ $\quad$

$\blacksquare$