Understanding Derivatives: From Limits to Jacobians

Derivative as instantaneous rate of change

In calculus, the derivative of a function at a point (when it exists) measures how sensitively the function’s output changes with respect to a change in its input at that point. Geometrically, it equals the slope of the tangent line to the function’s graph at the chosen input value, giving the best linear approximation near that point. This is why the derivative is often described as the instantaneous rate of change: it is the limiting value of the ratio of an “instantaneous” change in the dependent variable to the corresponding change in the independent variable. Formally, for a function f(x), differentiability at a means the limit of the difference quotient exists: lim(h→0) [f(a+h)−f(a)]/h. This limit represents the slope approached by secant lines as the two points on the graph get arbitrarily close. Differentiation is the process of finding derivatives, and the derivative can be written using several notations, including Leibniz notation (dy/dx) and prime notation (f′(x)). Higher-order derivatives arise by repeatedly differentiating and have physical interpretations, such as velocity (first derivative with respect to time) and acceleration (second derivative with respect to time). The idea extends beyond single-variable functions. For functions of several variables, the derivative becomes a linear transformation that provides the best linear approximation near a point; the Jacobian matrix represents this transformation in coordinates, and its entries are partial derivatives. The derivative also connects to concepts like continuity and differentiability: if a function is differentiable at a point, it must be continuous there, though continuity alone does not guarantee differentiability.

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Differentiability defined as a limit

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