leftD mathrme^-gammaleftw_T-1-pi_T-1 r_T-1right mathrme^-gamma pi_T-1 mu_T-1 mathrme^-gamma pi_T-1 sigma_T-1^2 2 mathrme^-gamma c_T-1 mathrme^-gamma deltaleftq_T-1right mathrme^-gamma q_T-1 alpha_T-1

使用乘法法则和链式法则，对 $\pi_{T-1}$ 求一阶导数得：

$$ \begin{aligned} \frac{\partial}{\partial \pi_{T-1}} & \left.D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2}\right]\ &= -\gamma D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} r_{T-1} \ &\quad -\gamma D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} \mu_{T-1} \ &\quad -\gamma D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} \frac{\sigma_{T-1}^2}{2} \ &\quad -\gamma D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} q_{T-1} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} \alpha_{T-1} \ &\quad -\gamma D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} q_{T-1} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} \beta_{T-1}^2 \ &= -\gamma \left[D \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} r_{T-1}\right. \ &\quad \left.+\mu_{T-1} \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2}\right. \ &\quad \left.-\frac{\sigma_{T-1}^2}{2} \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2}\right. \ &\quad \left.-q_{T-1} \mathrm{e}^{-\gamma\left(w_{T-1}-\pi_{T-1} r_{T-1}\right)} \mathrm{e}^{-\gamma \pi_{T-1} \mu_{T-1}} \mathrm{e}^{-\gamma \pi_{T-1} \sigma_{T-1}^2 / 2} \mathrm{e}^{-\gamma c_{T-1}} \mathrm{e}^{-\gamma \delta\left(q_{T-1}\right)} \mathrm{e}^{-\gamma q_{T-1} \alpha_{T-1}} \mathrm{e}^{-\gamma q_{T-1} \beta_{T-1}^2 / 2} \left(\alpha_{T-1}+\beta_{T-1}^2\right)\right] \end{aligned} $$

其中，第一行使用了乘法法则，第二到五行使用了链式法则