Differentiation / Chain rule - Splitting Logarithms

[binbagsss]

Homework Statement

Use the top line to get 1) and 2)

Homework Equations

above

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

$\frac{\partial}{\partial \tau} log (\eta(\frac{-1}{\tau})) = \frac{\partial}{\partial \tau} log (\eta(\tau)) \frac{\partial}{\partial \tau}(\frac{-1}{\tau})= \frac{1}{\tau^2} \frac{ \pi i}{12} E_2(\tau)$

 

[Buffu]

$(f(g(x)))^\prime = f^\prime(g(x))g^\prime(x)$

You applied the rule incorrectly,

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