Equivalence of Definitions of Derivative

Theorem

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

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

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

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 $\xi$.

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 $\xi$.

$\displaystyle f' \left({\xi}\right)$

$=$

$\displaystyle \lim_{h \mathop \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} h$

$\quad$

$\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 $

$=$

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

$\quad$

$\blacksquare$

2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 2.1$, Appendix $A$: Alternate Form of the Derivative