# Introduction of Derivatives

#### 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
lim_(h -> 0 ) [ f(a+h) - f(a)]/ h
provided this limit exists. Derivative of f (x) at a is denoted by f ′ (a).

Defination -
Suppose f  is a real valued function, the function defined by
lim_(h -> 0 ) [f(x + h) - f(x)]/h
wherever the limit exists is defined to be the derivative of f at x and is denoted by f ′ (x). This definition of derivative is also called the first principle of derivative.
Thus   f ' (x) = lim _( h -> 0) [f( x + h) - f(x)]/h

Clearly the domain of definition of f ′ (x) is wherever the above limit exists. There are different notations for derivative of a function. Sometimes f ′ (x) is denoted by d/(dx) f(x) or if y = f(x) , it is denoted by (dy)/(dx). This is referred to as derivative of f(x) or y with respect to x.
It is also denoted by D (f (x) ). Further, derivative of  f at x = a is also denoted by

d/(dx) f(x) |_a or (df)/(dx)|_aor even ((df)/(dx))_(x=a)

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