91Ó°ÊÓ

Calculus is a branch of mathematics that deals with change and motion. It explores how things change rather than the static nature of functions. There are two major components in calculus: differentiation and integration. Differentiation involves finding a derivative, which represents the rate of change of a quantity. Meanwhile, integration deals with finding the area under a curve.

In the given problem, we focus on differentiation. By analyzing how a function changes, calculus helps in understanding and predicting patterns, which is essential in fields like physics and engineering. Calculus adds depth to mathematical analysis, allowing us to approach problems involving changing quantities with precision.

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function changes at any point. It's like answering the question, "How is this changing?"

To differentiate a function, you generally apply rules like the power rule or the product rule, depending on the structure of the function. In our problem, we use differentiation to go from the original function to its first and then second derivatives. Finding the first derivative gives us insight into the function's velocity, while the second derivative can tell us about its acceleration. This second derivative helps us understand the concavity of the graph or the nature of curve changes over an interval.

The power rule is one of the most straightforward rules in differentiation, making it a helpful tool for calculus students. It states that if you have a term in the form of \( ax^n \), its derivative is \( nax^{n-1} \). This means you multiply the power by the coefficient and reduce the power by one.

In our exercise, the power rule was crucial for both the first and second derivatives. For example:

• The derivative of \( 4x^2 \) is \( 8x \), as we multiply 2 by 4.
• The derivative of \( 3x \) is \( 3 \), following the same rule since \( x \) is like \( x^1 \).

Applying the power rule simplifies the process of differentiating polynomials, making it easier to analyze functions and their changes. This rule simplifies many problems in calculus, allowing for quick and accurate calculations.