Türevlerdir

Türevlerdir is a Turkish phrase that translates roughly as “they are derivatives.” In Turkish mathematics, the noun türev denotes a derivative, and türevler is the plural form meaning derivatives. The form türevlerdir uses the plural noun with the copular suffix -dir to make a predicative statement, as in “Bu fonksiyonun türevlerdir” (these are derivatives of this function). The term appears primarily in grammatical or definitional contexts rather than as a distinct mathematical concept.

In calculus, a derivative measures the instantaneous rate of change of a function with respect to its variable. If f is defined on an interval around x, the derivative at x, denoted f′(x) or dy/dx, is the limit of the average rate of change as the increment tends to zero: f′(x) = lim h→0 [f(x+h) − f(x)]/h, provided the limit exists. Geometrically, f′(x) is the slope of the tangent line to the graph of f at x, and locally it provides a linear approximation: f(x+h) ≈ f(x) + f′(x) h.

Derivatives follow standard rules, such as the power rule, product rule, quotient rule, and chain rule, and can be extended to higher-order derivatives by applying differentiation repeatedly. Notation varies, with f′(x), dy/dx, and Df(x) all common. Derivatives have widespread applications across physics, engineering, economics, and beyond, including velocity as the derivative of position, marginal analysis, and optimization problems.

Historically, derivatives were developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, giving rise to the foundational methods of differential calculus. In Turkish textbooks, the concept is taught using the term türev, with türevler as its plural and phrases like türevlerdir used in explanatory statements or proofs.