Let
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,