# Derivatives

**Derivatives** are financial instruments that derive their value from underlying assets. Derivatives can be traded (bought and sold) in a manner similar to stock. Derivatives include futures, options and swaps.

The value and prices of derivatives depends on the fluctuating value of the underlying asset. But as derivatives become more important, their trading can influence the underlying asset pricing.

Derivatives range from the exotic to the mundane. Famously, Barclay's Bank trading in derivatives almost lead to their bankruptcy, but the purchase of a simple forward currency contract is also dealing in a derivative.

Derivatives can be traded on a financial exchange, or in an over-the-counter (OTC) market.

## Math

*(main article: Derivative)*

In calculus, a derivative is an operation on a function that shows the rate at which the output of the function is changing. The classic example of this is position with respect to time. Say some function f(t) is the position of an object at time t, then the derivative of f(t) would be the velocity of the object at time t, and further, the derivative of the velocity function would be the acceleration of the object at time t. The typical way of writing the derivative of a function f(t) would be f'(t).

A useful application of this is in physics.

The acceleration function a(x) is the derivative of the velocity function, v(x). V(x) is the derivative of the position or displacement function, p(x).
Example:
Displacement given by p(x) = 3x^{2}+6x+5
v(x) = p'(x) [shorthand for derivative of p(x) = 6x + 6
a(x) = v'(x) = 6
Thus, if x = 4, assuming the MKS units system for physics, the displacement is 77 m, the velocity is 30 m/s, and the acceleration is 6 m/s^{2}.