= Differential = A '''differential''' is a representation of an infinitesimally small change. <> ---- == Description == Given a function ''f(x)'', the differential of the function can be expressed in terms of the differential of ''x'': ''df = f' dx''. Note that while ''f' '' can also be expressed as ''df/dx'', that notation can mislead one to believing it is a trivial equality. The ''dx'' in the derivative cannot be 'cancelled out' by multiplying against the differential of ''x''. Note also that this equation holds only for infinitesimally small changes. For an observable change in ''x'', notated as ''Δx'', the corresponding change in ''f'' can only be [[Calculus/Linearization|approximated]] as ''Δf ≈ f' Δx''. Another way to view this is through limits: {{attachment:lim.svg}}. Clearly then the equation does not hold for non-zero ''Δx''. Given a multivariate function ''f(x, y, z)'', the '''total differential''' is expressed as ''df = f,,x,,dx + f,,y,,dy + f,,z,,dz'' where... * ''f,,x,, = ∂f/∂x'' * ''f,,y,, = ∂f/∂y'' * ''f,,z,, = ∂f/∂z'' A formulation for '''implicit differentials''' also follows. That is, given an surface ''f'' described by an equation like ''x^2^ + y^2^ = r'', the derivative of ''y'' with respect to ''x'' can be calculated as: {{attachment:imp.svg}} Beyond approximation, differentials also provide a framework for relating change. If the above function ''f'' were [[Calculus/ParametricEquation|parameterized]] in terms of time ''t'', then the equation can be divided by ''dt'' to give: {{attachment:diff.svg}} ---- CategoryRicottone