be the product of functions with derivatives . Find a formula for the derivative of , and prove that it is correct by mathematical induction.
Furthermore, show that
for those at which for any .
Claim: If , then
Proof. For the case , from the usual product rule we have
Thus, our claimed formula holds in this case. Assume then that it is true for some integer . Then, if we have,
Therefore, if the formula holds for then it also holds for ; thus, it holds for all positive integers
Next, we prove the formula for the quotient, .
Proof. Let be defined as above, and let be a point at which for any . Then,