91Ó°ÊÓ

The Power Rule is an essential tool in Calculus for finding derivatives, especially when dealing with polynomial functions. It's a simple rule that states: if you have a term in the form of \(t^n\), the derivative of this term, \(\frac{d}{dt}(t^n)\), is \(nt^{n-1}\). This means that you take the exponent of the term, multiply the entire term by this exponent, and reduce the exponent by one.
For instance, let's consider the term \(5.1t^2\) from our original function \(U(t) = 5.1t^2 - 1.1t\). Using the power rule here involves multiplying 5.1 by 2 (the original exponent), resulting in \(10.2t^{1}\).
This rule makes differentiating polynomial expressions straightforward, as you can apply it to each term independently. It's important to remember that constants, like \(-1.1t\), have their own specific treatment. The derivative of \(kt\), where \(k\) is a constant, is simply \(k\).

Finding derivatives is the process of determining the rate at which a function is changing at any given point. In our exercise, we are dealing with the function \(U(t) = 5.1t^2 - 1.1t\). The main aim is to find the derivative \(U'(t)\) using the power rule and then simplify the result.
Here's how it works step-by-step:

• Apply the power rule to each individual term of the function. For \(5.1t^2\), it gives us \(10.2t^{1}\).
• The constant derivative rule gives \(1.1\) for the term \(-1.1t\).

As a result, combining these calculations gives us the derivative \(U'(t) = 10.2t - 1.1\).
This process requires attention to details such as correctly applying the power rule and simplifying terms during the calculation process, leading to an accurate and simplified form of the derivative.

Once you have found the derivative function, evaluating it at a specific point is straightforward. For the function \(U(t)\), we've determined the derivative function is \(U'(t) = 10.2t - 1.1\).
To evaluate the derivative at \(t = 1\):

• Substitute \(t = 1\) into the derivative function \(U'(t)\).
• This gives us \(U'(1) = 10.2 \times 1 - 1.1\).

Complete the calculation to find \(U'(1) = 10.2 - 1.1 = 9.1\).
Evaluating derivatives is essential for understanding how a function behaves at specific points, indicating the slope of the tangent line to the curve at that particular point. This step is often crucial in applications of calculus where specific numerical values are required for further analysis or practical scenarios.