# Taking the partial derivative of the demand function

Define the demand function which maximizes x -> U(x) as:

$$\sum_{i=1}^np_i\zeta_i$$(p, I) = I

According to my textbook if I differentiate this with respect to $$p_j$$ I will obtain,

$$\zeta_j$$(p, I) + $$\sum_{i=1}^np_i\frac{\partial \zeta_i}{\partial p_j}$$(p,I) = 0

I don't understand how I get from the demand function to the output by differentiating w.r.t. $$p_j$$ do I have to use the chain rule, I tried that, I don't see why the function is equal to zero?

Thanks

• Yeah all that is being done is the chain rule and, since the RHS does not contain $p_j$, it becomes zero. Although this is more of a math question and would likely be more appropriate on math.se as it does not really contain economics per se. If I were to answer this question formally it would have no economics, only a math explanation of how to get from A to B – Brennan Apr 5 '20 at 16:44
• The $i$ and $j$ subscripts are missing in $\zeta$ (3 occurrences) – Bertrand Apr 5 '20 at 17:09