I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need to use the chain rule but I am not quite sure about implicit derivatives.
Your input is very much appreciated.
There is no problem around implicit differentiation.
Your formula is $$p_i=\frac{Aq_i^{\alpha-1}}K$$ Take logarithms of both sides to get $$\log(p_i)=\log(A)+(\alpha-1)\log(q_i)-\log(K)$$ Differentiate with respect to $q_i$ and get $$\frac 1{p_i}\frac{dp_i}{dq_i}=\frac{\alpha-1}{q_i}-\frac 1K\frac{dK}{dq_i}$$ Now, consider $$K=\sum_{j=1}^n q_j^\alpha$$ So $$\frac{dK}{dq_i}=\alpha q_i^{\alpha-1}$$ I am sure that you can take from here.