Fracdfdx

Fracdfdx is not a standard mathematical term. It is commonly encountered as a misformatted or abbreviated representation of the derivative df/dx, the instantaneous rate of change of a function f with respect to the variable x.

In its proper form, the derivative df/dx describes how f(x) changes as x changes. It applies to functions f: R -> R that are differentiable at a point x, meaning the limit of [f(x+h) − f(x)]/h as h approaches zero exists. The derivative is often denoted as df/dx, f′(x), or Df(x). For functions of multiple variables, the partial derivative ∂f/∂x expresses the rate of change of f with respect to x while holding other variables constant.

Common rules for computing df/dx include the power rule, product rule, quotient rule, and chain rule. Examples: if f(x) = x^3, then df/dx = 3x^2; if f(x) = sin(x), then df/dx = cos(x). The derivative also gives the slope of the tangent line to the graph of f at x and is central to linear approximation: f(x+h) ≈ f(x) + (df/dx) h for small h.

Differentials relate to the derivative via df = (df/dx) dx, and higher-order derivatives such as d^2f/dx^2 describe the curvature or acceleration of f. The concept generalizes to inverse functions, implicit differentiation, and in higher dimensions to gradient and directional derivatives. Related notions include df/dx notation, the total derivative, and partial derivatives ∂f/∂x.