There is nothing to derive. The formula is just the definition of the average failure rate. The only thing to be explained is a motivation for this choice. This is the following: Assume you have a huge set of elements of the same product, which can be destroyed by usage. Each element has its own lifetime. Statistical analysis gives you a number of active (=approptiately acting) elements, say N(t), which obviously decreases with time t>0. First we introduce the notion of the observed reliability function defined by the ratio R(t)= N(t)/N(0).
And now, assume you have the population of all elements still properly acting at age t1. And that you are interested in the fraction of the elements which will loose their ability to act properly till instant t2>t1. This would be N(t1) - N(t2) (= the number of elements destroyed within the time interval [t1,t2] ) divided by the initial number of elements (= N(t1) ), i.e. the ratio [N(t1) -N(t2)]/N(t1) which by simple division is equal to [R(t1) - R(t2)]/R(t1). Now it sufficies to remind that any intensities are calculated as changes per unit of time. Therefore, the observed intensity of changes of the reliability within the time interval is the result of division of [R(t1) -R(t2)]/R(t1) by (t2--t1), which is the quantity you are asking for. The name for this has been established as observed failure rate, since it counts (calculates) the rate of increase of observed failures related to the initial amount R(t1) of properly acting elements (i.e. at instant t1) per unit of time.
The theory of reliability has herited this notion and applied it to theoretical reliability, which is defined as the probability that an at random chosen element will live t units of time, which is R(t)=1-F(t), where F is the cdf (cumulative distribution function) of the random life time. Since the most useful functions are differentiable, the limit of the theoretical failure rate defined as above, i.e. by [R(t1)-R(t2)]/[R(t1)(t2-t1)] as t2 approches t1, is very useful. Obviously, this is the same as the derivative of -R(t+s))/R(t) with respect to s at s=t1. It is called instantaneous theoretical failure rate, and equals f(t)/R(t), where f(t) is the pdf (probability density function) of the life time.
In practical applications, many are forgetting adding the adjectives theoretical or observed, which may cause confusions.
Regards, Joachim
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