The derivative measures instantaneous rate of change. Formally, the derivative of \(f\) at a point \(x\) is defined as \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, \] provided this limit exists. An equivalent formulation focuses on a specific point \(a\): \[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}. \] These two forms are interchangeable and can be applied based on whichever is more convenient. The derivative exists at a point only when this limit is finite and well-defined; if the function has a corner, cusp, vertical tangent, or discontinuity at a point, the limit fails to exist and the derivative is undefined there.
Geometrically, the derivative \(f'(a)\) gives the slope of the tangent line to the graph of \(y = f(x)\) at the point \((a, f(a))\). The tangent line is the best linear approximation to the function near \(a\), meaning it matches the function's instantaneous direction at that point. Physically, if \(s(t)\) denotes the position of an object at time \(t\), then \(s'(t)\) represents the instantaneous velocity, and \(s''(t)\) the instantaneous acceleration. More broadly, the derivative captures the rate at which one quantity changes with respect to another.
An important logical relationship connects differentiability and continuity: if \(f\) is differentiable at a point, then \(f\) is necessarily continuous there as well. However, the converse fails. The classic counterexample is \(f(x) = |x|\) at \(x = 0\), where the function is continuous (since \(|0| = 0\) and the limit equals 0), but the sharp corner prevents the derivative from existing. This asymmetry between continuity and differentiability is one of the subtle but important ideas at the heart of calculus.