Definition Of The Derivatives

The derivatives    f'(x)  of the function   y=f(x)

at a given point   x  is defined by the equality

f'(x)=lim dx-->0 (dy/dx)

= lim dx-->0 [f(x+dx)-f(x)]/dx 

If this limit is finite, then the function

f(x)   is called differentiable at the point  x;  and

it is infallibly continuous at this point.

Geometrically, the value of derivative  f'(x)  represents

the slope of the line tangent to the graph of the function

y=f(x)   at the point   x   then;

+f'(x)=lim dx-->0 [f(x+dx)-f(x)]/dx 

is called the right side derivatives at the point   x   and

-f'(x)=lim dx-->-0 [f(x+dx)-f(x)]/dx

is called the left side derivatives at the point   x.

The necessary and the sufficient condition for the existence

 of the derivative   f'(x)  is the existence of both side derivatives,

also of the equality   -f'(x)=+f'(x).