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).