1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof by induction help.

  1. Sep 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove by induction on k that for all integers
    [tex] \frac{d}{dx} \prod_{i=1}^k f_i (x) = (\sum_{i=1}^k \frac{ \frac{d}{dx} f_i (x)}{f_i (x)} ) \prod_{i=1}^k f_i (x) [/tex]

    2. Relevant equations

    Product rule
    [tex] \frac{d}{dx} (uv) = u \frac{dv}{dx} + v \frac{du}{dx} [/tex]

    3. The attempt at a solution

    I am honestly not sure how to start this. It is the first one like this that I have tried.
  2. jcsd
  3. Sep 23, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    This problem is harder to write than it is to solve. Let's write "'" for d/dx and call P(k) the product of the f_i and S(k) the sum of the f'_i/f_i. Then what you have above is (P(k))'=S(k)P(k). Assume that's true. Now you want to prove (P(k+1))'=S(k+1)P(k+1), correct? P(k+1)=P(k)*f_(k+1) and S(k+1)=S(k)+f'_(k+1)/f_(k+1), also ok? Apply the product rule to P(k)*f_(k+1) and see if you can match the two sides up. And don't forget to prove the n=1 case to start the induction.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook