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In mathematics, the derivative of a function is one of the two central concepts of calculus. The derivative of a function at a point is a measure of the rate at which that function is changing as (one of) its independent variable changes. This corresponds to the slope of the tangent to the graph of the function at that point. If for instance the speed of a car is given as a function of time, then the derivative of this function describes the acceleration of the car as a function of time.

Suppose we wish to find the derivative of a suitably smooth function, f say, at the point x. If we increase x by a small amount, which we'll call Δx, we can calculate f(x + Δx). An approximation to the slope of the tangent to the curve is given by (f(x + Δx) - f(x)) / Δx, which is to say it is the change in f divided by the change in x. The smaller the value Δx, the better approximation.

Mathematically, we define the derivative, denoted f '(x), to be the mathematical limit of this ratio as Δx tends to zero. Functions which possess a derivative are said to be differentiable, and finding the derivative is also called differentiation. Instead of using the tedious limit definition in order to find the derivative at a single point, it is also possible to find the derivative function f ' (also written as df/dx) which records all derivatives of f at all points.

The success of calculus stems from the surprising fact that f '(x) can be easily computed from an expression for f(x) using a small number of algebraic rules:

  • Linearity: (a f + b g)' = a f ' + b g ' for all functions f and g and all real numbers a and b.
  • Power rule: If f(x) = xr with some real number r, then f '(x) = r xr-1.
  • Product rule: (f g)' = f ' g + f g' for all functions f and g.
  • Quotient rule: (f / g)' = (f ' g - f g') / g2
  • Chain rule: If f(x) = g(h(x)), then f '(x) = g'(h(x)) h'(x)

In addition, the derivatives of some special functions are useful to know:

  f(x)       f'(x)
  ex          ex
  ln(x)       1/x
  sin(x)      cos(x)
  cos(x)      -sin(x)
  tan(x)      sec2(x)
  cot(x)      -csc2(x)
  sec(x)      sec(x)tan(x)
  csc(x)      -csc(x)cot(x)

As an example, the derivative of f(x) = 2 x4 + sin(x2) - ln(x) ex + 7 is f '(x) = 8 x3 + 2x cos(x2) - 1/x ex - ln(x) ex.

Where a function depends on more than one variable, the concept of a partial derivative is used. Partial derivatives can be thought of informally as taking the derivative of the function with all but one variable held temporarily constant near a point. Partial derivatives are represented as ∂/∂x (where ∂ is a rounded 'd' known as the 'partial derivative symbol'). Mathematicians tend to speak the partial derivative symbol as 'der' rather than the 'dee' used for the standard derivative symbol, 'd'.

The concept of derivative can be extended to more general settings. The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.

In finance, a derivative is an investment, the value of which is based on, or derived from, an underlying security or commodity. Generally speaking, buying a derivative means buying the right to buy or sell the underlying security or commodity at some point in the future for a predetermined price. If the price of the underlying security or commodity moves into the right direction, the owner of the derivative makes money, otherwise they lose money. The potential loss or gain is much higher than if they had traded the underlying security or commodity directly.

Common examples of derivatives are options and futures.