2018-06 06/06/2018 ForsChem Research Reports 2018-06 (1 / 29) www.forschem.org Probability Density Functions of Derivatives of Random Variables Hugo Hernandez ForsChem Research, 050030 Medellin, Colombia hugo.hernandez@forschem.org doi: 10.13140/RG.2.2.23850.11204 Abstract Randomness emerges as the result of missing information. The presence of random variables in a model indicates that we are missing some critical information about the system. Such lack of knowledge propagates through functions involving random variables, including their derivatives. The purpose of this report is to show how to calculate the probability density function for the derivatives of random variables. As a general observation, the classical rules for the calculus of derivatives also apply to mathematical functions of random variables. The resulting probability density function can then be found by using the change of variables theorem. On the other hand, it is also possible to consider derivatives of independent variables. In this case, the probability density function of the derivative of an independent variable will always have a mean of zero, and it will be symmetrical around the mean. The multidimensional randomness of a system, captured through statistical or randomistic models can also be differentiated, and the probability density function of the derivative can be obtained analytically. In general, when normal random variables are involved, the resulting probability density function of the derivative can be approximated by a normal distribution. Keywords Change of Variable Theorem, Derivatives, Normal Distribution, Multidimensional Randomness, Multivariate Random Functions, Probability Density Functions, Random Variables 1. Introduction In previous reports [1,2], it was shown that the derivatives of random variables are also random variables. Therefore, derivatives of continuous random variables can also be described by probability density functions. The purpose of this report is to present the mathematical expressions for determining the probability density functions of derivatives of continuous random variables, and discuss some particular results.