Partial derivatives and multi-variable differentiation problem?
I added an image of the problem. I'm trying to figure out the first part (all of it really). My first thought was to solve s=(r-1)e^r to r and then derive dr/ds and it's partially working. I'm under the impression my answer for that is going to have to like (e^-r)/r since that's what w is equal to but? I feel like I'm going in the right direction but I'm stuck now. Any help would be great, thank you!
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s=(r-1)e^r --->ds/dr = re^r --->dr/ds = (1/r)e^(-r)
ðw/ðr = -(1+r)/(r^2 e^r) and ðr/ðu = -u/2
hence
ðw/ðu = (ðw/ðr)(ðr/ðu) = .... = (1/2)(1+(1/r))uw^2
similar for ðw/ðr