I claim that the three following definitions of the derivative are equivalent: $$ f’(x) = \begin{cases} \displaystyle \lim_{c \to x} \frac{f(c) - f(x)}{c-x} & \text{(A)} \\[10pt] \displaystyle \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} & \text{(B)} \\[10pt] \displaystyle \lim_{t \to 1} \frac{f(tx) - f(x)}{tx - x} & \text{(C)} \end{cases} $$ Most people who have taken a calculus course will be familiar with form B above, however there are contexts in which forms A and C are more convenient to use. Real analysis courses, for instance, often will prefer form A. In this post, I shall prove the equivalence claim.