Given is $F(x(p),y(p),z(p),t)=A(x(p),y(p),z(p))B(t)$
Is it correct to write
$\frac{dF}{dp}=[\frac{\partial A}{\partial x}\frac{dx}{dp}+\frac{\partial A}{\partial y}\frac{dy}{dp}+\frac{\partial A}{\partial z}\frac{dz}{dp}]B(t)$
Is it correct to also write
$\frac{\partial F}{\partial t}=A(x(p),y(p),z(p))\frac{dB}{dt}$
I was reading about the chain rule on Wikipedia and others but couldn't find a case with a similar product of functions.
EDIT: Also given is $B(t)=\frac{dp}{dt}$, thus $p$ depends on $t$, as shown by Fimpellizieri (see below)
If $p$ and $t$ are independent, meaning $\frac{d}{dt}p=\frac{d}{dp}t=0$, then yes, both are correct.
EDIT: If $p$ depends on $t$ via $\frac{d}{dt}p=B$, as indicated in the comments, then the chain rule (together with the product rule) for the second expression yields
$$\frac{\partial}{\partial t}F =\left(\frac{\partial }{\partial t}A\right)B+A\left(\frac{d }{d t}B\right)\\ =\left(\frac{\partial A}{\partial x}\frac{dx}{dp}\frac{dp}{dt} +\frac{\partial A}{\partial y}\frac{dy}{dp}\frac{dp}{dt} +\frac{\partial A}{\partial z}\frac{dz}{dp}\frac{dp}{dt}\right)B+A\cdot B'\\ =\left(\frac{\partial A}{\partial x}\frac{dx}{dp} +\frac{\partial A}{\partial y}\frac{dy}{dp} +\frac{\partial A}{\partial z}\frac{dz}{dp}\right)B^2+A\cdot B'$$