Bioreactor Substrate Yield Log (Chemical & Biomedical Engineering)

A biochemical engineer is monitoring an automated pilot bioreactor for synthetic therapeutic protein production. The automated process control system logs discrete sensor telemetry for microbial biomass concentration $X(t)$ (in $\text{g/L}$) and residual substrate glucose concentration $S(t)$ (in $\text{g/L}$), along with their respective rates of change:

1. The instantaneous biomass-to-substrate conversion ratio is defined by $Y(t) = \frac{X(t)}{S(t)}$. Write down the formal structural Quotient Rule formula for $\frac{dY}{dt}$.
2. Evaluate the instantaneous rate of change of the conversion ratio at peak exponential growth ($t = 4\text{ hr}$).
3. The total biochemical activity index is given by $A(t) = t \cdot X(t)$. Use the Product Rule to calculate $A'(6)$.