1. Average Rate of Change
1. Increment
   When the value of in a function changes from to , the function value changes from to .
1. The change in the value of , , is called the increment of .
2. The change in the value of , , is called the increment of .
   
   These are denoted symbolically as and , respectively. In other words:
   
   
   
   
2. Average Rate of Change
   When the value of in a function changes from to , the ratio of the increment of to the increment of is as follows:
   
   This is called the average rate of change of the function when changes from to .


2. Differentiation and the Geometrical Meaning of the Derivative
1. The average rate of change of the function when changes from to is:
   
   If the limit of this average rate of change as exists:
   
   this limit is called the instantaneous rate of change or the derivative of the function at , and it is denoted as .
2. The instantaneous rate of change or the derivative of the function at is:
   
3. Geometrical Meaning of the Derivative
   The derivative of the function at is the slope of the tangent line at the point on the curve .


3. Differentiability and Continuity
1. If the derivative of the function exists at , then the function is said to be differentiable at .
2. If a function is differentiable at , then is also continuous at .
   However, the converse is not true. In other words, a function can be continuous at but not differentiable at .