A derivative measures how fast a function's output reacts
The derivative of a single-variable function at a point is the limit of difference quotients as the step shrinks—an instantaneous rate of change. Leibniz wrote it as a ratio of differentials, while prime marks and other spellings still compete in classrooms.
For several variables the idea becomes a linear map that best approximates the function near a point after translation—the Jacobian viewpoint. The classic limit definition demands that for every epsilon a delta keeps the difference quotient within epsilon of the candidate slope L.
If that limit exists at every domain point, the derivative function itself can be studied, differentiated again, or fed into differential equations. Absolute-value inequalities in the epsilon–delta clause make the hand-waving of arbitrarily close into a checkable contract. Repeated differentiation is usually shown in Leibniz notation by attaching superscripts to the differentials.
Geometrically, the derivative at a point is the slope of the tangent line, and that tangent is the best straight-line approximation to the curve nearby. The difference quotient is the slope of a secant through two points on the graph; as the step shrinks, the two points close in and the secant slope settles toward the tangent's. For the squaring function the quotient simplifies to 2a plus h, so the derivative of squaring is doubling. Continuity alone is not enough: the absolute value function is continuous at zero yet has no derivative there. A separate route uses hyperreal numbers, an extension of the reals holding infinite quantities and their infinitesimal reciprocals; this approach, called nonstandard analysis, defines the derivative through infinitesimals and a standard part function that rounds each finite hyperreal to the nearest real number.
Source: Derivative
