Derivative multiple variables and summation

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I ran into a pretty nasty differential equation where we have the following: Differential in respects to k and I'm looking to seperate K here.

so what I recall from doing these 'in respects to' differentials, you consider other variables as constants and take the derivative like normal, while the summation can be put back into the formula after the differential is complete since the differential of a sum is the sum of the differential, right?

so I ended up with the completed differential but now I feel like this is incorrrect and I'm left wondering how to seperate K here. I'm not all that familiar with summations. Can somebody explain to me if I'm doing this right?

This formula is basically an equation for calculating a Mean Squared Error but we're interested in finding the slopes on the waves, something like that - It's for a friend and I'm not that confident in my ability to do these correctly, gave it a shot though.