Why do we write the stochastic differential equations in the following form $$dx(t)=f(x(t))dt+g(x(t))dW(t)$$ and we do not use: $$\frac{dx(t)}{dt}=f(x(t))+g(x(t))\xi(t)$$ where the white noise $\xi(t)=\frac{dW(t)}{dt}$, i.e. $\xi(t)$ is the generalized derivative of the brownian motion $W(t)$. However, $W(t)$ is not differentiable anywhere.
I am confused, any clarification would be appreciated.
I've already seen SDEs written as $$ \dot X(t) = f(X(t)) + g(X(t)) \, \dot W(t), \quad \quad (1) $$ so the notation you propose. Remark that both this notation and the more classical $$ \mathrm d X(t) = f(X(t)) \, \mathrm dt + g(X(t)) \, \mathrm dW(t), \quad \quad (2) $$ are actually only formally correct and are used as a common practical abbreviation. The rigorously correct way to write this would be employing the Itô integral, i.e., $$ X(t) = X_0 + \int_0^t f(X(s)) \, \mathrm ds +\int_0^t g(X(s)) \, \mathrm d W(s). \quad \quad (3) $$ Once it's established that (3) is the rigorous (correct) way to write an SDE down, the choice of writing (1) or (2) is arbitrary.