Solution: To find the maximum population, we first compute the derivative of $ P(t) $:

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### Why Mathematicians (and Data Analysts) Are Turning to Derivatives—What It Means for Understanding Growth Trends In a year marked by shifting demographics, economic recalibration, and advanced data-driven decision-making, a quiet but powerful tool is gaining attention: the derivative, particularly in the context of population modeling. For professionals across healthcare planning, urban development, and economic forecasting, understanding *when* population growth peaks—or stabilizes—plays a critical role in resource allocation and policy design. The method behind this insight lies in computing the derivative of the population function $ P(t) $, which reveals the rate at which a population changes over time, not just where it stands. This mathematical approach offers clear signals about growth patterns, enabling more accurate projections and informed strategic planning. Far from abstract, it speaks directly to real-world challenges in forecasting demand, managing infrastructure, and responding to social shifts. In the United States, where population shifts influence everything from housing markets to healthcare access, grasping these dynamics early can inform smarter, timely decisions—without requiring headlines designed to shock or sensationalize. The derivative, at its core, identifies moments of maximum expansion—when small changes in time produce the largest gains in population size. Rather than assuming growth continues linearly or steadily increases, this analytical tool captures the true rhythm of change, offering a sharper lens on population cycles. For planners, researchers, and individuals seeking clarity amid demographic uncertainty, this concept becomes a foundation for understanding long-term trends beyond surface-level data.