Diferentsiaalidel

Diferentsiaalidel, known more commonly in English as differentials, refers to a concept in calculus that represents an infinitesimal change in a variable. In simpler terms, it is a way to approximate the change in a function when its input changes by a very small amount.

The differential of a variable, denoted as $dx$ or $dy$, is not the same as the change in the variable, often denoted as $\Delta x$ or $\Delta y$. While $\Delta x$ represents a finite change, $dx$ is treated as an infinitesimally small change. This distinction is crucial in calculus, particularly in integration and the study of differential equations.

The relationship between the differential of a function and the differential of its independent variable is defined by the derivative. If $y = f(x)$, then the differential of $y$, denoted as $dy$, is given by $dy = f'(x)dx$. This equation shows that the differential of the function $y$ is equal to its derivative $f'(x)$ multiplied by the differential of the independent variable $x$.

Differentials are fundamental to understanding the integral as a sum of infinitely many infinitesimal rectangles, where the area of each rectangle can be approximated by $y \cdot dx$. They also play a key role in the chain rule for differentiation, where the differentials can be conceptually "cancelled out." The concept of differentials was an important step in the development of calculus, providing a rigorous framework for dealing with infinitesimals.