d2Udx2

d2Udx2 is the second derivative of a function U with respect to x. In formal notation, it is written d^2U/dx^2. The quantity describes how the slope dU/dx changes as x varies, and it is a measure of the curvature of U along the x-direction. If d^2U/dx^2 is positive at a point, the function is locally convex there; if negative, locally concave. If it is zero, higher-order derivatives determine the behavior, and the second-derivative test may be inconclusive for identifying extrema.

Common contexts: In single-variable calculus, d^2U/dx^2 appears in the second-derivative test for local extrema and in Taylor expansions as the coefficient of the (x-a)^2 term. In physics and engineering, it appears in equations of motion derived from potentials, where d^2U/dx^2 describes the stiffness or restoring force curvature near an equilibrium point. In multivariable problems, the second partial derivative with respect to x is denoted ∂^2U/∂x^2 and forms part of the Hessian matrix, which characterizes curvature in several directions.

Example: If U(x) = x^3 + 2x, then d^2U/dx^2 = 6x. When x>0 the second derivative is positive; when x<0 it is negative.

Notes: The term d2Udx2 can be used in plain-text contexts or programming languages that collapse formatting. It is not a distinct mathematical object beyond this derivative.

See also: derivative, second derivative test, Taylor series, Hessian matrix.