Rate of change is \( C'(t) = \frac{d}{dt}\left(10(t+2)^{-1}\right) = -10(t+2)^{-2} \).
["## Understanding the Rate of Change: Why ( C'(t) = -10(t+2)^{-2} ) Matters", "When analyzing how a quantity changes over time, one of the most powerful tools in calculus is the rate of change, formally expressed as the derivative ( C'(t) ). In calculus, the rate of change tells us exactly how fast ( C(t) ) is increasing or decreasing at any given moment — a fundamental concept in fields ranging from economics to physics.", "One classic example of computing this rate is calculating the derivative of a simply decreasing function, such as ( C(t) = 10(t+2)^{-1} ). This expression often arises when modeling phenomena like decay, diminishing returns, or cooling processes. But how do we derive its derivative, and why is the result ( C'(t) = -10(t+2)^{-2} )? Let’s explore this step by step.", "### What Does the Derivative Represent?", "The derivative ( C'(t) ) represents the instantaneous rate of change of ( C(t) ) with respect to time ( t ). For ( C(t) = \frac{10}{t+2} ), this derivative measures how quickly the value of ( C(t) ) shrinks as ( t ) increases — or equivalently, how fast the quantity decreases when modeled by this function.", "### Derivative Calculation", "Start with:\n[\nC(t) = 10(t+2)^{-1}\n]", "Apply the chain rule of differentiation, which is essential when dealing with composite functions. The chain rule states:\n[\n\frac{d}{dt}[f(g(t))] = f'(g(t)) \cdot g'(t)\n]", "Here:\n- Outer function: ( f(u) = 10u^{-1} )\n- Inner function: ( u = g(t) = t + 2 )", "First, differentiate the outer function with respect to ( u ):\n[\nf'(u) = \frac{d}{du}(10u^{-1}) = -10u^{-2}\n]", "Then differentiate the inner function:\n[\ng'(t) = \frac{d}{dt}(t + 2) = 1\n]", "Now, apply the chain rule:\n[\nC'(t) = f'(g(t)) \cdot g'(t) = -10(t+2)^{-2} \cdot 1 = -10(t+2)^{-2}\n]", "### Interpreting the Result", "The derivative:\n[\nC'(t) = -10(t+2)^{-2}\n]\nis negative for all ( t > -2 ), confirming that ( C(t) ) decreases as time progresses, as expected. The magnitude ( 10(t+2)^{-2} ) quantifies how rapidly the decrease slows down as ( t ) increases — approaching zero as ( t \ o \infty ), consistent with asymptotic behavior.", "### Why This Derivative Matters", "Understanding the rate of change like ( C'(t) ) helps in making informed predictions and decisions:\n- In finance, it models the diminishing growth of investments or depreciation.\n- In biology, it captures decay rates of substances or population decline.\n- In engineering, it quantifies cooling or pressure loss over time.", "By learning to compute and interpret derivatives such as ( C'(t) = -10(t+2)^{-2} ), individuals gain tools to analyze dynamic systems and respond effectively to changing environments.", "### Summary", "Calculating the rate of change using derivatives provides deep insight into how functions evolve. The expression ( C'(t) = -10(t+2)^{-2} ) precisely captures the shrinking trend of ( C(t) = \frac{10}{t+2} ), demonstrating calculus’ power to model real-world decay and guiding both theoretical understanding and practical applications.", "Mastering this concept strengthens analytical abilities across sciences, economics, and engineering — making it an essential skill for anyone working with dynamic systems."]
