Equivalence of Definitions of Derivative

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

$\blacksquare$


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