DfDx

dfdx is a shorthand used in calculus to denote the derivative of a function f with respect to the variable x. In standard notation this appears as df/dx or d f/dx, and it conveys the instantaneous rate at which f changes as x changes. If f is a function of a single variable x alone, df/dx is the total derivative of f with respect to x.

When f depends on several variables, the corresponding concept is the partial derivative, written ∂f/∂x, which measures how f changes with x while holding other variables constant. The total derivative can differ from a partial derivative if the other variables also depend on x, in which case the chain rule applies: df/dx = (∂f/∂x) + (∂f/∂y)(dy/dx) + …

Common examples: if f(x) = x^2, then d f/dx = 2x. If f(x) = sin(x^2), then df/dx = cos(x^2) · 2x. The second derivative, d^2f/dx^2, measures the curvature of f with respect to x, and is also written f''(x) in many contexts.

Applications span physics, engineering, economics, and beyond, where df/dx describes rates of change, slopes of graphs, or sensitivities. Numerically, derivatives may be approximated by finite differences, such as (f(x+h) − f(x))/h for small h.

Notes and caveats: df/dx assumes f is differentiable with respect to x. It is distinct from ∂f/∂x unless f depends only on x. In many contexts, d/dx denotes differentiation with respect to x, whereas gradient or Jacobian notation extends the concept to multivariable functions. The notation originates from Leibniz’s manuscript conventions.