dMdx

dM/dx is the derivative of a quantity M with respect to a variable x. It expresses the instantaneous rate at which M changes as x changes. If M is a differentiable function of x, dM/dx is defined by the limit lim_{h→0} [M(x+h) − M(x)]/h and is related to the differential by dM = (dM/dx) dx. The notation M′(x) is also common.

When M depends on several variables, partial derivatives ∂M/∂x measure the rate of change with respect to x while holding other variables fixed. The total derivative accounts for indirect dependencies via the chain rule: for a composition M = M(u(x)), dM/dx = (dM/du) (du/dx).

Examples: If M(x) = x^2, dM/dx = 2x. If M(x) = e^{2x}, dM/dx = 2 e^{2x}. For a composite function with u = x+1 and M(u) = u^3, dM/dx = (dM/du) (du/dx) = 3u^2 · 1 = 3(x+1)^2.

Applications and interpretation: In calculus, dM/dx gives the slope of the M versus x graph. In physics and engineering, it describes how a quantity such as magnetization, mass, or concentration changes with position or time. In economics or biology, similar rates of change are described by the same derivative concept. The derivative has units of M per unit of x.

Notes: Existence requires differentiability at x. If M is not differentiable at a point, dM/dx may not exist. For functions of several variables, ∂M/∂x denotes the partial rate of change, while the gradient ∇M collects all partial derivatives. Higher derivatives, such as d^2M/dx^2, quantify curvature.