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.