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Differentiation / Chain rule

  1. May 16, 2017 #1
    1. The problem statement, all variables and given/known data

    holdmeinurpantaloonies.png
    Use the top line to get 1) and 2)

    2. Relevant equations
    above
    3. The attempt at a solution
    So for 2) split the log up using ##log (AB)=log (A) + log (B) ## and this is simple enough

    I think I may be doing something stupid with 1) though. I have

    ##\frac{\partial}{\partial \tau} log (\eta(\frac{-1}{\tau})) = \frac{\partial}{\partial \tau} log (\eta(\tau)) \frac{\partial}{\partial \tau}(\frac{-1}{\tau})= \frac{1}{\tau^2} \frac{ \pi i}{12} E_2(\tau)##
     
  2. jcsd
  3. May 16, 2017 #2
    ##(f(g(x)))^\prime = f^\prime(g(x))g^\prime(x)##

    You applied the rule incorrectly,

    ##\displaystyle \frac{\partial}{\partial \tau} log (\eta(\frac{-1}{\tau})) = \frac{\partial}{\partial \tau} log (\eta(\color{red}{\frac{-1}{\tau}})) \frac{\partial}{\partial \tau}(\frac{-1}{\tau})= \frac{1}{\tau^2} \frac{ \pi i}{12} E_2(\color{red}{\frac{-1}{\tau}})##
     
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