Differential Calculus and its Applications | UGC NET Economics – Notes

The Concept of Derivative

Differential calculus is the branch of mathematics concerned with the study of rates of change, and it forms the single most indispensable mathematical tool in economic analysis, underlying marginal analysis, optimization theory, and elasticity measurement. The central concept of differential calculus is the derivative, which measures the instantaneous rate of change of a function with respect to a change in its independent variable.

Formally, the derivative of a function \(y = f(x)\) at a point (x) is defined as the limit of the ratio of the change in (y) to the change in (x), as the change in (x) approaches zero: $$f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x}$$ This limit, when it exists, is denoted variously as \(f'(x)\), \(\frac{dy}{dx}\), \(\frac{df}{dx}\), or \(D_x f(x)\). Geometrically, the derivative at a point represents the slope of the tangent line to the curve \(y = f(x)\) at that point. A function is said to be differentiable at a point if this limit exists there; differentiability at a point implies continuity at that point, though the converse is not true—a function may be continuous at a point yet fail to be differentiable there, as occurs at a sharp “kink” or corner in the graph.

The process of finding a derivative is called differentiation. Economically, the derivative is the mathematical embodiment of the concept of marginal: the marginal cost is the derivative of the total cost function with respect to output,\(\mathrm{MC} = \frac{dTC}{dQ}\); the marginal utility is the derivative of the total utility function with respect to quantity consumed, \(\mathrm{MU} = \frac{dTU}{dQ}\); and the marginal product of a factor is the derivative of the total product (output) function with respect to that factor’s employment, \(\mathrm{MP}_L = \frac{dQ}{dL}\).