Solution:** The rate of change of efficiency is given by the derivative \( E'(t) \). Compute \( E'(t) \):

["Understanding the Rate of Change of Efficiency: Computing the Derivative ( E'(t) )", "In dynamic systems across engineering, economics, and scientific modeling, the concept of efficiency is central to evaluating performance over time. Knowing not just the efficiency value but how quickly it changes—its rate of change—offers vital insights for optimization, forecasting, and decision-making. A key mathematical tool in this analysis is the derivative ( E'(t) ), representing the instantaneous rate of change of efficiency ( E(t) ) with respect to time.", "### What Is the Rate of Change of Efficiency?", "The rate of change of efficiency describes how efficiently a system improves or deteriorates at a specific moment. Whether tracking a manufacturing process, a learning curve, or an energy conversion unit, ( E'(t) ) reveals whether performance is accelerating, stabilizing, or declining.", "Mathematically, the rate of change is given by the derivative:", "[\nE'(t) = \frac{dE}{dt}\n]", "This expression captures the sensitivity of efficiency to time, enabling precise adjustments and predictive modeling.", "### How to Compute ( E'(t) ): Step-by-Step Guide", "Computing ( E'(t) ) requires identifying the functional form of efficiency ( E(t) ), which depends on the specific system you're analyzing. Below is a structured approach:", "1. Define the Efficiency Function ( E(t) ):\n Begin by expressing efficiency as a function of time or another independent variable. For example, ( E(t) = at - bt^2 ) might model a decaying efficiency under wear and tear.", "2. Apply Differentiation Rules:\n Use standard calculus rules to differentiate ( E(t) ):\n - Power Rule: If ( E(t) = kt^n ), then ( E'(t) = nkt^{n-1} )\n - Sum Rule: ( (f + g)' = f' + g' )\n - Product Rule: ( (fg)' = f'g + fg' )\n - Chain Rule: ( \frac{d}{dt}[f(g(t))] = f'(g(t)) \cdot g'(t) )", "3. Simplify the Expression:\n After differentiation, algebraically simplify ( E'(t) ) to obtain a clear, interpretable formula.", "### Example: Computing ( E'(t) ) for a Quadratic Efficiency Model", "Suppose efficiency over time is modeled by:\n[\nE(t) = 100 - 5t - 2t^2\n]\nApplying differentiation term-by-term:\n[\nE'(t) = \frac{d}{dt}[100] - \frac{d}{dt}[5t] - \frac{d}{dt}[2t^2] = 0 - 5 - 4t = -5 - 4t\n]", "This derivative reveals efficiency decreases linearly over time, and the rate of decline accelerates (( -4t ) becomes steeper as ( t ) increases).", "### Practical Applications", "- Manufacturing: Monitor efficiency loss in machinery to schedule maintenance.\n- Learning Efficiency: Track knowledge acquisition curves to optimize study schedules.\n- Energy Systems: Analyze real-time performance drops in solar panels under variable weather.", "### Conclusion", "Computing ( E'(t) ) transforms a static efficiency value into a dynamic performance indicator, empowering data-driven decision-making. By mastering differentiation techniques applied to ( E(t) ), professionals and researchers gain actionable insights into system behavior over time.", "Keywords:\nefficiency derivative, rate of change of efficiency, computing ( E'(t) ), derivative of efficiency function, optimal system performance, calculus in efficiency analysis, real-time efficiency monitoring", "---", "Understanding and computing ( E'(t) ) is not just a mathematical exercise—it’s a powerful strategy to harness time-dependent efficiency data and drive sustainable improvements across industries. Whether you're an engineer, economist, or data scientist, leveraging the rate of change provides a clearer lens through which to view and enhance system performance."]