Example:
            Let
(a) Find the average rate of change of  with respect to over the interval.
(b) Find the instantaneous rate of change of with respect to  at the point .

Solution:

(a) For Average Rate of Change:
            We have
           
            Put
           
            Again Put
           
            The average rate of change over the interval  is
           
(b) For Instantaneous Rate of Change:
                        We have
           
            Put
           
            Now, putting  then
           
            The instantaneous rate of change at the point  is
           
            .

Example:
            A particle moves on a line away from its initial position so that after  seconds it is  feet from its initial position.
(a) Find the average velocity of the particle over the interval.
(b) Find the instantaneous velocity at .

Solution:

(a) For Average Velocity:
            We have
           
            Put
           
            Again Put
           
            The average velocity over the interval  is
           
(b) For Instantaneous Velocity:
            We have
           
            Put
           
            Now putting
           
            The instantaneous velocity at  is
           
                  
                  

Example:
            Use the formula for the area of a circle to find,
(a) the average rate of which the area of a circle changes with  as the radius increases form  to .
(b) the instantaneous rate at which the area of a circle changes with when .

Solution:

(a) For Average Velocity:
            We have
           
            Put
           
            Again Put
           
            The average rate of change from  to  is
           
(b) For Instantaneous Rate of Change:
            We have
           
            Put
           
            Now putting
           
            The instantaneous rate of change at the point   is
           
            .