Derivoidaan

Derivoidaan is the Finnish passive present form of the verb derivoida, meaning to differentiate or to compute the derivative. It is used to describe the process of finding the derivative of a function with respect to a variable.

In calculus, the derivative of a function f at a point x0 represents the instantaneous rate of change of f with respect to x, and geometrically it corresponds to the slope of the tangent line to the graph of f at x0. For a function defined on an interval I, the derivative is written as f'(x) or df/dx, provided the limit that defines the derivative exists: f'(x0) = lim_{h→0} [f(x0+h) − f(x0)]/h.

Notation varies. The derivative can be written as df/dx, f'(x), or Df(x). The operation is local, meaning it depends on the behavior of the function near the point of differentiation.

Key rules apply when derivoidaan functions. The sum rule states that the derivative of a sum is the sum of the derivatives. The product rule says (uv)' = u'v + uv'. The quotient rule gives (u/v)' = (u'v − uv')/v^2, and the chain rule states (f(g(x)))' = f'(g(x)) g'(x). Higher-order derivatives are denoted d^2f/dx^2, d^3f/dx^3, and so on.

Examples: derivoidaan f(x) = x^2 to obtain f'(x) = 2x. Derivoidaan f(x) = sin x to obtain f'(x) = cos x.

In contexts with several variables, derivatives include partial derivatives, and the term derivoidaan may be used in general language, though osittaisderivoidaan is used for clarity when specifying the variable of differentiation.