Equivalence of Definitions of the 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.

Proof

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

$\blacksquare$

Sources

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