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,