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 \(z_{it}\).

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

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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