Alternatively, perhaps the function is \( C(t) = 10e^{-0.1t} \), then \( C' = -1e^{-0.1t} \), set to -1: then \( e^{-0.1t} = 1 \Rightarrow t = 0 \).
["Alternative Derivation of Derivative for Exponential Decay Function", "When analyzing models of exponential decay commonly used in physics, finance, and biology, the function ( C(t) = 10e^{-0.1t} ) frequently appears. Understanding its rate of change—its derivative—provides deep insight into how quantities decrease over time. This article explores the alternative yet equivalent method to compute ( C'(t) ) and verify when ( C'(t) = -1 ).", "The given function models a decay process where ( C(t) ) decreases over time with time constant 10 and decay coefficient 0.1. The mathematical expression for the derivative is found using the chain rule:", "[\nC(t) = 10e^{-0.1t}\n]", "Applying the chain rule, we differentiate:", "[\nC'(t) = 10 \cdot (-0.1) e^{-0.1t} = -1 e^{-0.1t}\n]", "This confirms the derivative ( C'(t) = -1e^{-0.1t} ), as expected.", "Now, to find when the derivative equals (-1), set up the equation:", "[\n-1e^{-0.1t} = -1\n]", "Dividing both sides by (-1):", "[\ne^{-0.1t} = 1\n]", "To solve for ( t ), take the natural logarithm of both sides:", "[\n\ln(e^{-0.1t}) = \ln(1)\n]", "Using logarithmic identities:", "[\n-0.1t = 0\n]", "Thus,", "[\nt = 0\n]", "This reveals a key insight: the rate of decay is maximal at time ( t = 0 ), and the instantaneous rate ( C'(t) ) drops to (-1) precisely when time begins, reflecting the steepest downward slope. This result aligns with intuition—decay rates are fastest at the start of the process.", "In summary, analyzing ( C(t) = 10e^{-0.1t} ) and computing its derivative alternatively confirms ( C'(t) = -e^{-0.1t} ), and demonstrates that ( C'(t) = -1 ) only when ( t = 0 ). This analytical approach not only verifies the derivative numerically but also illuminates the physical meaning of decay rates over time.", "For anyone studying differential equations, calculus, or decay modeling, mastering such derivations strengthens understanding of dynamic systems governed by exponential behavior.", "---", "Keywords: ( C(t) = 10e^{-0.1t} ), derivative, ( C'(t) ), exponential decay, chain rule, ( e^{-0.1t} = 1 ), ( t = 0 ), calculus explanation, decay rate analysis."]
