The derivatives

*of the function*

**f'(x)**

**y=f(x)**at a given point

*is defined by the equality*

**x**

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

**= lim dx-->0 [f(x+dx)-f(x)]/dx**If this limit is finite, then the function

*is called differentiable at the point*

**f(x)***and*

**x;**it is infallibly continuous at this point.

Geometrically, the value of derivative

*represents*

**f'(x)**the slope of the line tangent to the graph of the function

*at the point*

**y=f(x)***then;*

**x**

**+f'(x)=lim dx-->0 [f(x+dx)-f(x)]/dx**is called the right side derivatives at the point

*and*

**x**

**-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

*is the existence of both side derivatives,*

**f'(x)**also of the equality

**-f'(x)=+f'(x).**