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The Power Rule is a basic yet essential technique in calculus used when differentiating polynomials. When you encounter a function like U(t) = 5.1t^2, the Power Rule simplifies the process of finding its derivative. How does it work? When you have a term in the form at^n, where a is a constant and n is a positive integer, the derivative according to the Power Rule is n * a * t^(n-1).

In the exercise, applying the Power Rule to the term 5.1t^2 leads to a derivative of 2 * 5.1 * t^(2-1), which simplifies to 10.2t. This step is crucial for understanding how to find the rate of change at any given point of a polynomial function. Additionally, this example highlights the need for attention to both the exponent and the coefficient during differentiation.

Dealing with constant terms in differentiation might seem confusing at first, but it's pretty straightforward: the derivative of a constant is always zero. This is because a constant term does not change regardless of the value of the variable; hence, its rate of change is zero. In our problem, the term 5.1 does not depend on t, making its derivative 0.

When evaluating a function like U(t) = 5.1t^2 + 5.1, remember to ignore the constant when computing the derivative because it does not contribute to the slope of the function at any point. This concept is helpful to eliminate unnecessary steps and focus on the terms that affect the rate of change in a function.

Evaluating derivatives refers to the process of finding the value of a function's derivative at a specific point. Once you've used rules like the Power Rule to find the derivative in a general form, the next step is to substitute the variable with the desired value. In this case, to find \(U'(3)\), you plug in \(t = 3\) into the derived function U'(t) = 10.2t.

The result, \(U'(3) = 10.2 * 3 = 30.6\), tells you that at \(t = 3\), the slope of the tangent line to the curve or the rate of change of the function \(U(t)\) is 30.6. This practical application of derivatives can be employed for various purposes, such as determining speed at a particular time for physics problems or calculating marginal cost in economics at a specific production level.