Derivatiivse

Derivatiivse is a term used in mathematics, specifically in calculus, to refer to the rate at which a function changes. It is essentially the instantaneous slope of a curve at a particular point. The concept of the derivative is fundamental to understanding motion, optimization, and many other scientific and engineering applications.

The derivative of a function y = f(x) with respect to x is denoted by f'(x) or dy/dx. It is defined as the limit of the difference quotient as the change in x approaches zero: lim (h->0) [f(x+h) - f(x)] / h. This limit represents the slope of the tangent line to the graph of the function at the point (x, f(x)).

Derivatives have numerous applications. In physics, the derivative of position with respect to time gives velocity, and the derivative of velocity with respect to time gives acceleration. In economics, derivatives can be used to find the marginal cost or marginal revenue of a product. In optimization problems, derivatives help locate maximum and minimum values of a function.

The process of finding a derivative is called differentiation. There are specific rules and formulas for differentiating various types of functions, such as polynomials, trigonometric functions, exponential functions, and logarithmic functions. Understanding these rules allows for efficient calculation of derivatives. The inverse operation of differentiation is called integration, which is used to find the area under a curve.