derivative
or differential coefficient, n. 1. (for a function f(x) at the argument x) the limit of the difference quotient 
as the increment Δx tends to 0. For functions of a single variable, if the left- and right-hand limits exist and are equal, it is the gradient of the curve at x, and is the limit of the gradient of the chord joining the points (x, f(x)) and (x + Δ x, f(x + Δ x)), as shown.
The function of x defined as this limit for each argument x is the first derivative of y = f(x); it is the rate of change of the value of the function with respect to the independent variable, and is written
dy/dx, f'(x), or Dx f(x),
while the ratio of differences of which this is the limit is written δy/δx. The process of extracting the derivative is called differentiation. For example, the first derivative of axn is anxn-1. The second derivative is the first derivative of the first derivative, and is written
d2y/dx2, f''(x), D2 f(x), or Dxx f(x).
These definitions are readily extended to functions of several variables; see partial derivative. 2. the Gateaux derivative or Frèchet derivative of a vector space mapping.
as the increment Δx tends to 0. For functions of a single variable, if the left- and right-hand limits exist and are equal, it is the gradient of the curve at x, and is the limit of the gradient of the chord joining the points (x, f(x)) and (x + Δ x, f(x + Δ x)), as shown.dy/dx, f'(x), or Dx f(x),
while the ratio of differences of which this is the limit is written δy/δx. The process of extracting the derivative is called differentiation. For example, the first derivative of axn is anxn-1. The second derivative is the first derivative of the first derivative, and is written
d2y/dx2, f''(x), D2 f(x), or Dxx f(x).
These definitions are readily extended to functions of several variables; see partial derivative. 2. the Gateaux derivative or Frèchet derivative of a vector space mapping.