The geometrical and physical interpretation of fractional order derivatives for a general class of functions

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non-monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order (Formula presented.), including non-monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function (Formula presented.), with (Formula presented.) being a positive real number, the value (Formula presented.) is a turning point when (Formula presented.). However, for (Formula presented.), the turning point shifts to the left of point (Formula presented.) and shifts to the right of point (Formula presented.) for (Formula presented.) We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis.