second derivative — from formula anatomyThe second derivative measures how the rate of change of a quantity is itself changing. If the first derivative of a function tells you its slope or velocity, the second derivative tells you the slope's slope — the acceleration of the quantity. In physics, it appears everywhere: Newton's second law relates force to the second derivative of position (acceleration), and the wave equation ties spatial and temporal second derivatives together.
The concept emerged with the development of calculus in the late 17th century, primarily through the work of Isaac Newton and Gottfried Wilhelm Leibniz, who formalized derivatives of any order.
For a function f(x), the first derivative f'(x) gives the instantaneous rate of change. The second derivative f''(x) is the derivative of f'(x), computed by applying the same limiting process again. Geometrically, it indicates concavity: positive means the function curves upward, negative downward, zero marks an inflection point. In motion, if x(t) is position, then x''(t) is acceleration — how quickly velocity changes.
Used in engineering for vibration analysis (second derivative of displacement gives acceleration), in economics for marginal returns, and in physics for deriving equations of motion and wave propagation.
The second derivative of a straight line is always zero, which is why constant-velocity motion feels smooth — no acceleration.
| f''(x) | variable | Second derivative of function f at x — second derivative per metre squared |
| f(x+h) | variable | Function value at x+h — function value |
| f(x-h) | variable | Function value at x-h — function value |
| f(x) | variable | Function value at x — function value |
| h | variable | Step size — spacing metre |
| \approx | operator | Approximately equal to — approximation |
| \psi | variable | wave function — wave function |
| x | variable | position coordinate — position metre |
| k | variable | wave number — wave number per metre |
| V(x) | variable | potential function — potential energy joule |
| \frac{d^2}{dx^2} | operator | second derivative with respect to position — second derivative |
| ψ | variable | wave function — wave function |
| x | variable | position coordinate — position metre |
| V(x) | variable | potential energy function — potential energy joule |
| E | variable | energy eigenvalue — energy joule |
| d²/dx² | operator | second derivative with respect to position — second derivative |
| V | variable | effective potential — potential energy joule |
| f | variable | function whose curvature defines the potential |
| x | variable | independent variable — position metre |
| d²/dx² | operator | second derivative with respect to x — second derivative |
| z | variable | scale factor or redshift depending on context — scale factor |
| ν | variable | Hankel index — hankel index |
| τ | variable | conformal time — conformal time second |
| '' | operator | second derivative with respect to the independent variable (here, conformal time) — second derivative |
