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:

Time $t\text{ (hr)}$	$X(t)\text{ (g/L)}$	$S(t)\text{ (g/L)}$	$X'(t)\text{ (g/(L}\cdot\text{hr))}$	$S'(t)\text{ (g/(L}\cdot\text{hr))}$
2	$1.5$	$18.0$	$0.4$	$-2.1$
4	$2.8$	$12.5$	$0.9$	$-3.5$
6	$5.2$	$5.0$	$1.2$	$-4.0$
8	$6.8$	$1.2$	$0.3$	$-1.8$

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 $\frac{dY}{dt}$ .
Evaluate the instantaneous rate of change of the conversion ratio at peak exponential growth ( $t = 4\text{ hr}$ ).
The total biochemical activity index is given by $A(t) = t \cdot X(t)$ . Use the Product Rule to calculate $A'(6)$ .