It is well known that the calculation of an accurate approximate derivative f ′ ( x ) f\prime (x) of a nontabular function f ( x ) f(x) on a finite-precision computer by the formula d ( h ) = ( f ( x + h ) − f ( x − h ) ) / 2 h d(h) = (f(x + h) - f(x - h))/2h is a delicate task. If h is too large, truncation errors cause poor answers, while if h is too small, cancellation and other "rounding" errors cause poor answers. We will show that by using simple results on the nature of the asymptotic convergence of d ( h ) d(h) to f ′ f\prime , a reliable numerical method can be obtained which can yield efficiently the theoretical maximum number of accurate digits for the given machine precision.