Complicated partial derivative

[RobP]

I have the following function:

[MATH]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)[/MATH]
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.,

[MATH]\frac{\partial y_{it}}{\partial z_{it}} = ?[/MATH]
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 \(z_{it}\).

 

[RobP]

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

[MATH]\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}}.[/MATH]
Computing the above is not difficult.