How would one go about partially differentiating the following with respect to L;
z = $\frac{T}{L}\Psi_{z} - \frac{T}{L}\sum_{n=1}^N X(n) sinh[\frac{2 \pi n}{L}(h + z)] cos(\frac{2 \pi n}{L}x)$
I used the product rule to arrive at;
$ \frac{\delta z}{\delta L} = -\frac{T}{L^{2}} + \frac{T}{L}[\sum_{n=1}^N X(n)[\frac{2\pi n}{L^{2}}x sin (\frac{2\pi n}{L}x) sinh(\frac{2\pi n}{L}(h + z)) - \frac{2\pi n}{L^{2}}(h + z) cos (\frac{2\pi n}{L}x)cosh(\frac{2\pi n}{L}(h + z))] - \frac{T}{L^{2}}[\sum_{n=1}^N X(n)cos(\frac{2\pi n}{L}x)sinh(\frac{2\pi n}{L}(h + z))] $
However I'm not sure if this is the correct procedure for differentiating when a summation series is involved. On a brief Google I saw mentions of power series, albeit not involving product rules, so it may be just a gap of knowledge on my own part here.
Any assistance on this would be much appreciated, even if it's just a link to some worked examples of a similar nature!
The derivative of the (finite) sum is the sum of the derivatives. This is the usual sum rule from Calc I. If the series were infinite you would need to check a few things before interchanging derivative with sum.