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This is about a differentiating problem, see details in main box

I don’t see how to get from (9) to (10).

By differentiating wrt wi, I get the beta part, but i don’t get the second part. Can someone explain this? Thanks.

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Answer 1195

I’m going to answer this as a mathematical question as the economic meaning of the variables is unclear without more context.

The second part of the differentiation is an application of the basic rule for differentiating powers: $d(ax^n)/dx = nax^{n-1}$

In this case:

$a = q[w_1w_2…w_m]^{1/m}$ where the product in square brackets includes all the w’s except $w_i$.

$x = w_i$

$n = 1/m$

Differentiation of $w_i^{1/m}$ with respect to $w_i$ yields $(1/m)w_i^{(1/m)-1}$ which equals $w_i^{1/m}/mw_i$.

When this is multiplied by $q[w_1w_2…w_m]^{1/m}$, the $w_i$ term in the numerator can be included in the product in the square brackets producing the full set of w’s $w_1$ to $w_m$. Hence the differential can be expressed as $(q/mw_i)[w_1w_2…w_m]^{1/m}$.

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