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 $x = \xi + h$ | $\quad$ | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{x \mathop \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}\) | $\quad$ | $\quad$ |
$\blacksquare$
Sources
- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 2.1$, Appendix $A$: Alternate Form of the Derivative