The problem involves differentiating the product of n functions, denoted as g = f1 * f2 * ... * fn, and proving the resulting rule using mathematical induction. The original poster aims to derive a formula for g'(x)/g(x) under the condition that none of the function values are zero at point x.
Some participants have shared their attempts at deriving the differentiation rule and have begun to outline the induction proof. There is acknowledgment of the complexity of the problem, and while some progress has been made, there is no explicit consensus on the correctness of the approaches taken.
Participants note the need for a base case in the induction proof and the requirement to show that if the statement holds for n = k, it must also hold for n = k + 1. There is an emphasis on ensuring that the functions involved do not equal zero at the points of interest.
Show us what you did. For an induction proof, you need to establish a base case, and then assume that the statement is true when n = k. Then you need to show that when the statement for n = k is true, the statement for n = k + 1 must also be true.ben.tien said:Homework Statement
: Let f1,...fn be n functions having derivatives f'1...f'n. Develop a rule for differentiating the product g = f1***fn and prove it by mathematical induction. Show that for those points x, where none of the function values f1(x),...fn(x) are zero, we have g'(x)/g(x) = (f'1(x)/f1(x))+...(f'n(x))/(fn(x))
Homework Equations
product rule: (f1*f2) = (f'1*f2 + f1*f'2)
The Attempt at a Solution
: So I used the associativity property to bunch up the n functions into n/2 functions : (f'1*f2 + f1*f'2)...(f'n-1*fn + fn-1*f1n) and that's where I got stuck.