exponentiel

Exponential, or exponentiel in French, designates a function or phenomenon characterized by growth or decay proportional to its current value. In mathematics, a standard exponential function takes the form f(x) = a^x, where the base a satisfies a > 0 and a ≠ 1. The natural exponential, denoted exp(x), is e^x, where e ≈ 2.71828 is the base of the natural logarithms.

Properties: The function is monotone increasing if a > 1 and monotone decreasing if 0 < a < 1. Derivatives satisfy d/dx a^x = a^x ln a, while d/dx exp(x) = exp(x). Integrals follow ∫ a^x dx = a^x / ln a + C and ∫ exp(x) dx = exp(x) + C. The function is invertible on its domain, with inverse log base a: log_a(x) = ln x / ln a. The exponential with base e is its standard form in calculus, and exp(x) can be expanded as a power series exp(x) = ∑_{n=0}^∞ x^n / n!.

Domain and asymptotics: For a > 0 and a ≠ 1, a^x is defined for all real x. As x → ∞, a^x → ∞ if a > 1, while a^x → 0 as x → -∞; the behavior reverses for 0 < a < 1.

Applications: Exponentials model growth and decay in biology, physics, and chemistry, as well as financial processes such as compound interest. Discrete growth follows A = P(1 + r)^t, while continuous compounding uses A = P e^{rt}. Exponential models also appear in differential equations, population dynamics, and statistics, where they describe processes that change at a rate proportional to their current value.