Delta of an option is defined as ratio of change in price of call option to change in price of underlying securities. If, $c_t$ is call option price at time $t$ and $S_t$ is the price of underlying securities, then the delta of call option is:
$$\Delta(t)=\frac{\partial c_t}{\partial S_t}$$
If change in $dS_t$ ie $(S_{t+dt} -S_t)$ is random, which by definition is true, then delta ($\Delta (t)$, computed at time t) must be random variable (instead of known constant) as it involve $dS_t$ in denominator. But under the Black-Scholes model, the delta of European of Call option (which is written as $N(d_1)$ or $\Phi (d_2)$) is deterministic variable (ie known with certainty) at time $t$.
I want to know why delta of a call option is deterministic quantity, why not it is random variable? If possible, please provide both logical reasoning and formal derivation.