I have the following dynamic programming problem:
$$ \max_{C_t,H_t} \ V_{t}^{w}= \left\{ \left[ (C_t^w)^\upsilon(1-H_t)^{1-\upsilon} \right]^\rho + \beta \left[ \omega V_{t+1}^w + (1-\omega)V_{t+1}^r \right]^\rho \right\}^{1/\rho}, $$ subject to $$ A_t^w = R_{t-1}A_{t-1}^w + W_t H_t - T_t - C_t^w. $$
I have the FOC wrt $C_t$ as: $$ 0 = \frac{1}{\rho}V_t^w(V_t^w)^{-1} \left\{ \rho\upsilon(C_t^w)^{\rho\upsilon-1}(1-H_t)^{(1-\upsilon)\rho}+\rho\beta \left[ \omega V_{t+1}^w +(1-\omega)V_{t+1}^r \right]^{\rho-1} \left[ \omega\frac{\partial V_{t+1}^w}{\partial A_{t}^{w}} + (1-\omega)\frac{\partial V_{t+1}^r}{\partial A_t^r} \right] (-1) \right\} $$
Assuming that $\partial V_{t+1}^r / \partial H_t=0$, is this the correct derivative for $\partial V_t / \partial H_t$?
$$ 0 = \frac{1}{\rho}V_t^w(V_t^w)^{-1} \left\{ (1-\upsilon)\rho(C_t^w)^{\upsilon\rho}(1-H_t)^{(1-\upsilon)\rho-1}(-1) + \rho\beta \left[ \omega V_{t+1}^w +(1-\omega)V_{t+1}^r \right]^{\rho-1} \left[ \omega\frac{\partial V_{t+1}^w}{\partial A_t^w} \right] W_t. \right\} $$
Have I applied chain rule correctly for this problem? More specifically, are the last terms, $-1$ and $W_t$, for the respective FOCs, correct?