Derivatives. Geometric interpretation

Derivatives are a fundamental concept in differential calculus. Simply put, the derivative of a function at a given point provides us with information about how that function changes in the vicinity of that point.

The derivative of a function is defined as the limit of the incremental ratio when the interval of change in the independent variable tends to zero. This incremental ratio is calculated by dividing the difference of the values of the function at two nearby points by the difference of the values of the independent variable at those same points. The derivative is usually denoted as f'(x) or dy/dx, and can be interpreted geometrically as the slope of the tangent line to the curve at that point.

Derivatives have many applications in various fields, such as physics, economics, engineering, and data science. In physics, for example, derivatives are used to describe the velocity and acceleration of a moving object. In economics, they are used to analyze rates of change in variables such as production, consumption and income. In engineering, derivatives are fundamental to study the behavior of dynamic systems and design efficient controllers.

There are rules and properties that facilitate the calculation of derivatives. These rules make it possible to find the derivative of a composite function, the product of two functions or a function raised to a power.

In addition to ordinary derivatives, there are also partial derivatives, which are used in the calculation of functions of several variables. Partial derivatives measure the rate of change of a function in relation to each of its independent variables, keeping all other variables constant.

The online derivative simulations on this page are a very useful tool to deepen your knowledge in this field of mathematics. Make the most of them!