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This thesis consists of five parts. The first section contains definitions and theorems that are needed in following sections. These mostly regard Euler’s gamma and beta functions. This includes the domain of Euler’s gamma function, the relationship between the two functions and the gamma function’s limit representation. The second section consists of three chapters and provides Grünwald-Letnikov’s, Riemann-Liouville’s and Caputo’s approaches to fractional derivatives. Each chapter includes the definitions, the reasoning behind those definitions and some of the more important properties of those definitions. The third section concerns the relationships between the three previously stated definitions of fractional derivatives. Firstly, it explores the connection between Riemann-Liouville’s and Caputo’s fractional differential operators, and secondly, the relationship between Riemann-Liouville’s and Grünwald-Letnikov’s definitions of fractional derivatives. The fourth section gives examples on how to find fractional derivatives of some power functions based on the three different definitions explored in this paper. The example functions are (x − a)b, √x, x2, x3 and the constant function f(x) = c. The final section gives examples of the applications of fractional derivatives. It primarily focuses on how to solve Abel’s integral equation and equations that are equivalent to Abel’s integral equation using fractional order derivatives and how to model some special cases of diffusion.