Your team member suggested to use LSTM instead of RNN for sentiment analysis project, where LSTM cell is defined as

$i_t=(W^{(i)}{x}_t+U^{(i)}{h}_{t-1})$

$f_t=(W^{(f)}{x}_t+U^{(f)}{h}_{t-1})$

$o_t=(W^{(o)}{x}_t+U^{(o)}{h}_{t-1})$

tilde$(c{)}_t=tanh(W^{(c)}{x}_t+U^{(c)}{h}_{t-1})$

$c_t=f_t$@$c_{t-1}+i_t$@tilde$(c{)}_t$

$h_t=o_t$@$tanh(c_t)$

i$)$ What should be the values of input gate and forget gate if you want the cell $c_t$ to sum its inputs over time.?

If the forget gate is set to $1$ for a cell dimension and the input gate set to $0,$ then the information of that cell is preserved indefinitely.

ii$)$ Derive the gradient del$\frac{J}{d}el{U}^{(c)}$ in terms of the following gradients: del$\frac{h_t}{d}el{h}_{t-1},$del$\frac{h_{t-1}}{d}el{U}^{(c)},$ del$\frac{J}{d}el{h}_t,$del$\frac{c_t}{d}el{U}^{(c)},$del$\frac{c_{t-1}}{d}el{U}^{(c)},$del$\frac{c_t}{d}el{c}_{t-1},$del$\frac{h_t}{d}el{c}_t,$ and del$\frac{h_t}{d}el{o}_t.$ Do not calculate the individual gradients. You can leave the answer in form of chain rule.

Solve the $($ii$)$ question full explanation.