(a) Use the limit definition of the derivative to find for .
(b) Use the limit definition of the derivative to find for .
(c) Use the limit definition of the derivative to find for . (Hint: . Apply this rule to within the limit definition.)
(d) Based on your work in (a), (b), and (c), what do you conjecture is the derivative of ? Of ?
(e) Conjecture a formula for the derivative of that holds for any positive integer . That is, given where is a positive integer, what do you think is the formula for ?
You can earn partial credit on this problem.