Tiderivater

Tiderivater, or time derivatives, are derivatives of quantities with respect to time. In mathematics and physics, the time derivative of a quantity q(t) measures how q changes as time progresses. The most common notation is the derivative with respect to t, written as dq/dt. In physics, time derivatives are often shown using a dot notation: ẋ for dx/dt, ẍ for d^2x/dt^2, and higher derivatives as x^(n)(t) or with successive dots.

A fundamental example is kinematics: for a particle with position x(t), the velocity is v = dx/dt = ẋ, and the acceleration is a = dv/dt = ẍ. The third derivative x⃛ = d^3x/dt^3 is sometimes called the jerk, and higher derivatives have analogous names in some contexts.

Time differentiation is a linear operator, obeying rules such as the product rule d/dt(fg) = f' g + f g' and the chain rule d/dt f(g(t)) = f'(g(t)) g'(t). The derivative of a constant is zero. When a function depends only on time, the time derivative is the ordinary derivative with respect to t. If the function depends on time and other variables, the appropriate concept is the partial derivative with respect to time, ∂/∂t, or, in moving media, the substantial derivative D/Dt.

Applications of tiderivater span many fields. In physics and engineering, they describe motion and dynamic systems via differential equations, such as m dv/dt = F. In control theory, system evolution is written as dx/dt = f(x,u,t). In signal processing, time derivatives relate to changes in signals and their frequency content. Time derivatives thus provide a core tool for modeling change over time across disciplines.