dNdx

dN/dx, often written in ASCII as dNdx, denotes the derivative of a function N with respect to the variable x. It measures the instantaneous rate at which N changes as x changes, assuming N is differentiable with respect to x.

In calculus, dN/dx represents a total derivative when N depends on x directly. If N also depends on other variables that themselves depend on x, the appropriate concept is the total derivative, which accounts for both direct and indirect effects. When N depends on multiple independent variables, the partial derivative ∂N/∂x is used to describe the rate of change of N with respect to x while holding other variables constant. The differential form dN = (dN/dx) dx expresses the infinitesimal change in N for an infinitesimal change in x.

Calculation proceeds via limits or standard differentiation rules. For example, if N(x) = x^3, then dN/dx = 3x^2. If N(x) = a x^2 + b x + c, the derivative is 2a x + b. Differentiation rules, such as the chain rule, product rule, and quotient rule, extend to a wide range of functions.

Interpretation and applications: dN/dx is used to assess slopes of graphs and to model rates of change in physics, chemistry, biology, economics, and engineering. The units of dN/dx are the units of N per unit of x. It is a foundational concept in differential calculus, underpinning methods such as Taylor expansions, optimization, and the formulation of differential equations.