There is no doubt that contemporary differentiation (including Frechet derivative) had its origin in the calculus of variations. In this chapter, we delve into the foundational concepts of functional minimization within the framework of variational calculus, focusing on their applications in image processing. We begin by defining the Gâteaux derivative, a key tool for understanding variations in functionals, which paves the way for formulating optimization problems. Following this, we explore the first variation and the gradient of functionals, deriving the associated Euler-Lagrange equations for various fundamental functionals. Finally, we introduce variational methods in the context of minimal surface theory, where the surface area is minimized, offering deeper insights into both geometric problems and their practical applications in image processing. This chapter equips the reader with critical tools for formulating and solving variational problems.