Complicated partial derivative

RobP

New member
Joined
Jun 29, 2019
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2
I have the following function:

\(\displaystyle y_{it} = ln \left(a_{1}+ \sum_{j=2}^{J}\frac{a_{j}}{e^{q_{ji}+b_{jit}}}\right)+\frac{1}{r}\sum_{j=2}^{J}a_{j}q_{ji}-ln\left(a_{1}+\sum_{j=2}^{J}(a_{j}e^{-b_{jit}})\right)\)

where \(e\) above is the natural exponent. Furthermore, \(q_{ji}\) is a composite function, i.e., \(q_{ji}(z_{it})\). I want to calculate the partial derivative of \(y_{it}\) with respect to \(z_{it}\), i.e.,

\(\displaystyle \frac{\partial y_{it}}{\partial z_{it}} = ?\)

I need the expression as an input to a larger problem that I am working on. Any help is highly appreciated.

ps. You can indicate the derivative of \(z_{it}\) as \(z^{\prime}_{it}\) in the solution since I do not provide the expression for \(z_{it}\).
 
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RobP

New member
Joined
Jun 29, 2019
Messages
2
I think I figured this out. It is evident that \(y_{it}= f(q_{2i}, q_{3i}, \cdots, q_{Ji})\) and \(q_{2i}= q_{2i}(z_{it})\), \(q_{3i}= q_{3i}(z_{it})\), \(\cdots\), \(q_{Ji}= q_{Ji}(z_{it})\). Applying the chain rule, we have

\(\displaystyle \frac{dy_{it}}{dz_{it}}=\frac{\partial y_{it}}{\partial q_{2i}}\cdot\frac{dq_{2i}}{dz_{it}}+ \frac{\partial y_{it}}{\partial q_{3i}}\cdot\frac{dq_{3i}}{dz_{it}}+\cdots + \frac{\partial y_{it}}{\partial q_{Ji}}\cdot\frac{dq_{Ji}}{dz_{it}}.\)

Computing the above is not difficult.
 
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