Deriv

Deriv is commonly used as shorthand for derivative, a central concept in calculus that measures the instantaneous rate at which a function changes. For a real-valued function f defined on an interval, the derivative at a point x is defined by the limit 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 the point (x, f(x)). It can also be interpreted as the velocity or rate of change of a quantity whose value is f(t).

Notations frequently used for derivatives include f'(x), df/dx, dy/dx, and Df(x). Differentiation obeys several rules, such as linearity (D(af+bg) = a Df + b Dg), the chain rule, the product rule, and the quotient rule. Common elementary derivatives are d/dx x^n = n x^{n-1}, d/dx e^x = e^x, d/dx sin x = cos x, and d/dx cos x = -sin x.

Higher-order derivatives are obtained by differentiating again; the second derivative f''(x) relates to concavity and, in physics, to acceleration. A function may have derivatives at some points but not others; non-differentiable points include corners, cusps, vertical tangents, or discontinuities.

Applications of derivatives span mathematics, science, and engineering. They are used to model rates of change, optimize functions (finding maxima and minima), analyze motion, and study physical phenomena such as velocity and acceleration. In multivariable contexts, partial derivatives and gradients generalize the concept to functions of several variables. The derivative concept emerged in the 17th century, developed independently by Newton and Leibniz, and remains foundational in analysis and applied disciplines.