partial derivative with respect to vector how to solve the following partial derivative with respect to vector of probabilities p?
$$\vec c=[c_1,...,c_L], c_i\in\{0,1\}$$
$$\vec p=[p_1,...,p_L], p_i\in[0,1]$$
$$\frac \partial {\partial \vec p} \prod_{d=1}^{L}p_d^{c_d}(1-p_d)^{1-c_d}$$
Thanks
 A: I may contribute partially.
Let $$ f(\vec{c}) = \left( p_{1}^{c_{1}} (1-p_{1})^{1-c_{1}}\right)\left( p_{2}{c_{2}} (1-p_{2})^{1-c_{2}}\right) \cdots \left( p_{L}^{c_{L}} (1-p_{L})^{1-c_{L}}\right)  $$
$$ \frac{ \partial  f}{ \partial \vec{p}}  =  \frac{ \partial }{ \partial \vec{p}} \left( p_{1}^{c_{1}} (1-p_{1})^{1-c_{1}}\right)\left( p_{2}^{c_{2}} (1-p_{2})^{1-c_{2}}\right) \cdots \left( p_{L}^{c_{L}} (1-p_{L})^{1-c_{L}}\right)   $$
Using the definition of
$$ \frac{ \partial f(\vec{c}) }{ \partial \vec{p}} = \lim_{h \rightarrow 0} \frac{ f(c_{1} + hp_{1}, c_{2} + hp_{2}, \cdots,c_{L}+hp_{L}) - f(c_{1}, c_{2}, \cdots,c_{L})}{h}  $$
and by simplifying 
$$ f(c_{1} + hp_{1}, c_{2} + hp_{2}, \cdots,c_{L}+hp_{L}) - f(c_{1}, c_{2}, \cdots,c_{L})= \left[ \left( p_{1}^{c_{1}} (1-p_{1})^{1-c_{1}}\right)\left( p_{2}^{c_{2}} (1-p_{2})^{1-c_{2}}\right) \cdots \left( p_{L}^{c_{L}} (1-p_{L})^{1-c_{L}}\right)   \right] \left[\left( \frac{p_{1}}{1-p_{1}} \right)^{hp_{1}} \cdots \left( \frac{p_{L}}{1-p_{L}} \right)^{hp_{L}} - 1 \right]   = \left[ \left( p_{1}^{c_{1}} (1-p_{1})^{1-c_{1}}\right)\left( p_{2}^{c_{2}} (1-p_{2})^{1-c_{2}}\right) \cdots \left( p_{L}^{c_{L}} (1-p_{L})^{1-c_{L}}\right)  - 1 \right] \left[ \Pi \left( \frac{p_{d}}{1-p_{d}} \right)^{p_{d} h} - 1 \right]  $$
we can get
$$ \frac{ \partial f(\vec{c}) }{ \partial \vec{p}} = \left[ \Pi  p_{d}^{c_{d}} (1-p_{d})^{1-c_{d}} \right] \lim_{h \rightarrow 0} \frac{ \left[ \Pi \left( \frac{p_{d}}{1-p_{d}} \right)^{p_{d}}  \right]^{h} -1}{h} $$
You may continue from here..?
