Reaction Rate Kinematics and pH Buffering (Chemical Engineering)

A chemical process engineer is designing a dynamic automated buffering system to neutralize an acidic byproduct stream in an industrial chemical plant. The active concentration of hydronium ions $H^+$ (measured in moles per liter, $\text{M}$) over an operating period follows a combined product decay profile:

where $t$ is the elapsed processing time in minutes ($t \ge 0$).

The industrial acidity sensor tracks the effluent stream using the standard logarithmic pH scale:

To automate neutralizing alkaline buffer injection, the control computer tracks the derivative $\frac{d(\text{pH})}{dt}$.

1. Deconstruct the global composite structure to express $\frac{d(\text{pH})}{dt}$ using the Chain Rule.
2. Differentiate the outer logarithmic sensor scale $\frac{d(\text{pH})}{dH^+}$ using the general logarithm derivative rule.
3. Differentiate the hydronium concentration $H^+(t)$ with respect to time $t$ using the Product Rule and the Exponential Chain Rule.
4. Combine the layers to find the symbolic expression for $\frac{d(\text{pH})}{dt}$ in terms of $t$, and evaluate the rate of change at exactly $t = 4\text{ minutes}$.