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A derivative in mathematics represents a rate of change of a function. When we talk about the motion of an object, the derivative helps us track how the position changes over time. In the case of our exercise, the distance function, \( s(t) = 16t^2 \), represents how far a baseball falls over a period of time. By taking the derivative of this function, we are essentially asking: "How fast is the position changing at any given moment?" This is known as the velocity. To compute the derivative, we apply calculus principles to transform our distance equation into one that expresses velocity. This is an essential tool in calculus, as it provides a way to measure instantaneous rates of change, like speed, at any specific point in time.

The power rule is a fundamental technique in calculus used for differentiating functions that are powers of a variable. It tells us how to easily compute the derivative of functions like \( t^n \). For a function \( s(t) = 16t^2 \), the power rule states that \( \frac{d}{dt}[t^n] = nt^{n-1} \). This means, to find the derivative of \( t^2 \), you multiply by the exponent 2, and then decrease the exponent by one, giving you \( 2t^{1} = 2t \). For our distance function, multiplying by the coefficient 16 gives us the derivative \( 32t \). Using the power rule allows for swift differentiation, which is crucial for efficiently solving physics and calculus problems.

The distance function in our context describes how far the baseball falls over time. It is expressed as \( s(t) = 16t^2 \), where \( s \) is the distance in feet and \( t \) is the time in seconds. This specific form, \( at^2 \), relates to freely falling objects beginning at rest under the influence of gravity, where the coefficient typically reflects half of the gravitational acceleration, which is approximately 32 feet per second squared on Earth. Thus, the 16 in our equation highlights the gravitational effect halved, making it apt for scenarios like dropping objects to track their descent over time. Understanding this function is critical as it provides a foundation upon which we apply calculus techniques to gain deeper insights like differentiating to find velocity.

Velocity calculation involves determining how quickly something moves, considering direction. Calculus provides tools to find the instantaneous velocity of an object by taking the derivative of its distance function with respect to time. In our instance, the distance function is \( s(t) = 16t^2 \). By differentiating, we obtained \( v(t) = 32t \), which represents the velocity function. To find the instant velocity at a specific time, say \( t = 3 \), we substitute into the equation: \( v(3) = 32 \times 3 = 96 \) feet per second. This calculation is crucial, as it tells us the speed of the baseball at that very moment, helping us understand dynamic motion in real-time. So, the next time you watch a ball drop, remember this simple yet powerful calculation.