Taking a derivative of a function wrt time, when t is not itself a variable (just a subscript).

This is in an economics setting, where I have a standard Cobb Douglas function that puts output in terms of technology, labour and capital:

$$Y_t=A_tK_t^{\alpha}L_t^{1-\alpha}$$

Function to be differentiated

I can take the partial derivative of the A, K and L terms easily, but how do I obtain dY/dt? My notes say to use product rule, but the t is only a subscript not a variable.

Logarithmic differentiation gives

$$\frac{\dot{Y_t}}{Y_t}=\frac{\dot{A_t}}{A_t}+\alpha \frac{\dot{K_t}}{K_t}+(1-\alpha)\frac{\dot{L_t}}{L_t}\\ \implies\dot{Y_t}=Y_t\left(\frac{\dot{A_t}}{A_t}+\alpha \frac{\dot{K_t}}{K_t}+(1-\alpha)\frac{\dot{L_t}}{L_t}\right).$$

Also, $t$ is a variable; the use of subscripts is just notation.