If ( ) , , , mx f x nx m nx m < ≤ ≤ > . For what integers m and n does both f x and f x exist? . Derivatives We have seen in the Section .
, that by knowing the position of a body at various time intervals it is possible to find the rate at which the position of the body is changing. It is of very general interest to know a certain parameter at various instants of time and try to finding the rate at which it is changing. There are several real life situations where such a process needs to be carried out. For instance, people maintaining a reservoir need to know when will a reservoir overflow knowing the depth of the water at several instances of time, Rocket Scientists need to compute the precise velocity with which the satellite needs to be shot out from the rocket knowing the height of the rocket at various times.
Financial institutions need to predict the changes in the value of a particular stock knowing its present value. In these, and many such cases it is desirable to know how a particular parameter is changing with respect to some other parameter. The heart of the matter is derivative of a function at a given point in its domain of definition. MATHEMATICS Definition Suppose f is a real valued function and a is a point in its domain of definition.
The derivative of f at a is defined by f a f a provided this limit exists. Derivative of f ( x ) at a is denoted by f ′ ( a ) . Observe that f ′ ( a ) quantifies the change in f ( x ) at a with respect to x. Example Find the derivative at x = of the function f ( x ) = x.
Solution We have f ′ ( ) = f f lim3 The derivative of the function x at x =