Derivaatan

Derivaatan is a term associated with the concept of the derivative. In Finnish mathematics, derivaatta is the word for derivative, and derivaatan is the genitive form meaning “of the derivative.” The derivative measures the instantaneous rate of change of a function with respect to its variable.

For a real-valued function f defined on an interval, the derivative at a point a is f'(a) = lim_{h→0} [f(a+h) − f(a)] / h, provided the limit exists. It can be denoted in various notations, such as df/dx or Df(a). Derivatives generalize to functions of several variables, where the derivative at a point is represented by the Jacobian matrix for a vector-valued function or by the gradient for a scalar-valued function.

Geometrically, the derivative at a point is the slope of the tangent line to the graph at that point and provides the best linear approximation: f(a+h) ≈ f(a) + f'(a) h for small h. The derivative plays a central role in many rules, including linearity (derivative of a sum is the sum of derivatives), the product rule (fg)' = f'g + fg', the quotient rule, and the chain rule for compositions (f∘g)'(x) = f'(g(x)) g'(x).

In multiple dimensions, differentiability implies the existence of a linear map that best approximates the change in the function, captured by the Jacobian. The gradient vector contains the partial derivatives and points in the direction of steepest ascent. Derivatives underpin many applications, from physics and engineering to economics and optimization, and are foundational in calculus and analysis.