I am studying thermodynamics. I am trapped at this step. I know it's a chemistry but no one really helped me there.
This is a symbolic expression where: $E = f(T, V)$ (or Energy is a function of T and V)
Then the derivative was calculated as this:
$dE = (\frac{dE}{dT})_{V} ~dT + (\frac{dE}{dV})_{T}~dV $
I just don't know how he get this expression. I know in a partial derivative you need to keep one variable constant, but how they are summed up like this? and why he equates with dE and not dE/dt?
You are right when you say that you need to keep one variable constant. But, since $E$ depends on two variables, you need to consider the contribution of both variables to know completely how $E$ changes with those variables. The way to consider the contribution of each variable is simply summing then, and that is why the expression has two terms.
Furthermore, the expression provides a formula to calculate $dE$, i.e., an infinitesimal variation of $E$. An infinitesimal variation of $E$ must be proportional to the infinitesimal variations of $T$ and $V$, i.e., $dE = A dT + B dV$. If all those variables change with, say, time $t$, you can divide the expression by $dt$ (tip: don't do this if there are mathematicians close to you!) to obtain $$ \frac{dE}{dt} = \frac{\partial E}{\partial V} \frac{dV}{dt} + \frac{\partial E}{\partial T} \frac{dT}{dt}, $$ i.e., the rate of change of $E$ is proportional to the rate of change of $V$ and $T$.