In mathematics, the term function derivative is the basic concept of differential calculus. Many problems in the natural, technical, and social sciences can be solved by function derivative. In this work, the notion of function derivative and its application in geometry are described in more details. At the very beginning, a historical overview of the development of the differential calculus is given. The most important mathematicians and their contribution to the development of differential calculus, which developed together with integral calculus under the common name of infinitesimal calculus, are listed. The notion of function derivative using the problem of linear approximation is introduced. The derivative of a function at a point is defined and the derivative rules for elementary functions, composition of functions and inverse functions are given. In the work are given the basic theorems of differential calculus and their geometric interpretation. After that, the local theory of plane or spatial curves is presented. It is described how the tangent to a plane curve is determined, the curvature and torsion of spatial curves are defined, and the fundamental theorem for curves is presented. At the very end, examples of optimization problems in geometry are given, which are solved by applying a function derivative.