While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led me to think that maybe I got something wrong about full time-derivative, which is as follows:
$$ \frac{d\,f(q_1,q_2,\dots,q_N,p_1,p_2,\dots,p_N,t)}{dt}=\sum_{j=1}^{N}\left( \frac{\partial f}{\partial q_j} \underbrace{\frac{\partial q_j}{\partial t}}_{=\dot{q_j}=\frac{\partial H}{\partial p_j}}+ \frac{\partial f}{\partial p_j} \underbrace{\frac{\partial p_j}{\partial t}}_{=\dot{p_j}=-\frac{\partial H}{\partial q_j}} \right)+\frac{\partial f}{\partial t} =\{f,H\}+\frac{\partial f}{\partial t} $$
Now if for example I use function $f$ as $q_k$, by saying: $f(q_1,q_2,\dots,q_N,p_1,p_2,\dots,p_N,t)=q_k$ then I get following:
$$ \frac{dq_k}{dt}=\sum_{j=1}^{N}\left( \underbrace{\frac{\partial q_k}{\partial q_j}}_{=\delta_{kj}} \frac{\partial H}{\partial p_j}- \underbrace{\frac{\partial q_k}{\partial p_j}}_{=0} \frac{\partial H}{\partial q_j} \right)+\frac{\partial q_k}{\partial t} =\{q_k,H\}+\frac{\partial q_k}{\partial t} = \underbrace{\frac{\partial H}{\partial p_k}}_{=\dot{q_k}} + \underbrace{\frac{\partial q_k}{\partial t}}_{\stackrel{?}{=}\dot{q_k}}\stackrel{?}{=}2\dot{q_k} $$
If I cross out the $\frac{\partial q_k}{\partial t}$ on both sides of equation $\{q_k,H\}+\frac{\partial q_k}{\partial t}=\frac{\partial H}{\partial p_k}+\frac{\partial q_k}{\partial t}$ then I recover the Hamilton's equation $\dot{q_k}=\{q_k,H\}$. But if I don't do this, and go forward with the $2\dot{q_k}$ that appears at the end I get this very surprising identity:
$$ \frac{dq_k}{dt}\stackrel{?}{=}2\dot{q_k} $$
or written another way:
$$ \frac{dq_k}{dt}\stackrel{?}{=}2\frac{\partial q_k}{\partial t} $$
My question is following: is this really true? If not, then what have I done wrong? If yes, then why it is not mentioned anywhere in the textbooks - wouldn't that be some other way to find the $\dot{q_k}$?
Note:The derivation goes exactly the same for $\dot{p_k}$.
EDIT: Thanks to your answers I marked now the wrong equalities with $\stackrel{?}{=}$ since I hate to see incorrect math typed out. But I still wanted to preserve this question as how it was written at first. Also in the first equation the full time derivative for $\dot{q_j}$ and $\dot{p_j}$ should be used like this:
$$ \frac{df}{dt}=\sum_{j=1}^{N}\left( \frac{\partial f}{\partial q_j} \underbrace{\frac{d q_j}{d t}}_{=\dot{q_j}=\frac{\partial H}{\partial p_j}}+ \frac{\partial f}{\partial p_j} \underbrace{\frac{d p_j}{d t}}_{=\dot{p_j}=-\frac{\partial H}{\partial q_j}} \right)+\frac{\partial f}{\partial t} $$