Method for deriving an Exponential Moving Average?

Question

I have a formula for an exponentially weighted moving average function defined recursively as:

$S_t = a*Y_t+(1-a)*S_{t-1}$

Where:

• $a\in (0,1)\cap \mathbb{Q}$
• $t$ represents time
• $Y_t$ is the value at a time period $t$
• $S_t$ is the value of the EMA at any time period t.

I am trying to find a generally applicable solution for the derivative of $S_t$ in regards to $dt$.

Any assistance would be most appreciated!