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The rate of change is a concept that describes how one quantity changes in relation to another. When you find the rate of change of a function, you are essentially discovering how quickly the dependent variable responds to changes in the independent variable. For example, in the exercise, we are looking at the function \(y = 2x^3 + 2\) and are asked to find how \(y\) changes as \(x\) changes when \(x = 2\).

In mathematics, the rate of change is often represented by the derivative. The derivative of a function at a given point provides the slope of the tangent line at that point, symbolizing the instantaneous rate of change. In this case, when \(x = 2\), the rate at which \(y\) changes with respect to \(x\) is the derivative of \(y\) evaluated at that point, which tells us how \(y\) is increasing or decreasing as \(x\) increments by a tiny amount.

Therefore, the rate of change at a specific point offers a precise snapshot of the behavior of the function at that instant, making it a powerful tool for understanding trends and making predictions based on functional relationships.

One of the most efficient techniques in differentiation is the power rule. It is particularly useful when dealing with polynomial functions. The power rule states that if you have a term in the form of \(x^n\), its derivative will be \(nx^{n-1}\), where \(n\) is a constant exponent. This method simplifies the differentiation process by offering a straightforward, consistent formula to apply.

In the given problem, the function \(y = 2x^3 + 2\) can be differentiated using the power rule. By treating \(2x^3\) as a separate term, you apply the power rule: multiplying the coefficient 2 by the derivative of \(x^3\), which is \(3x^2\), resulting in \(6x^2\).

• Differentiate \(x^3\) to get \(3x^2\)
• Multiply by the coefficient 2 to obtain \(6x^2\)

The constant \(2\) differentiates to zero, as constants don't change regardless of \(x\). Thus, the derivative \(\frac{dy}{dx} = 6x^2\) provides the rate of change of the function \(y\) concerning \(x\).

Utilizing the power rule simplifies the calculation, turning what could be a cumbersome task into a quick and efficient process. This makes it a fundamental tool in single-variable calculus.

Single variable calculus is a branch of mathematics focusing on functions of a single independent variable. This stream is often seen as the foundation for more advanced calculus concepts. Its primary focus is on understanding concepts of limits, derivatives, and integrals with respect to one variable.

In the context of the given exercise, how a simple polynomial function such as \(y = 2x^3 + 2\) behaves as \(x\) changes demonstrates prime applications of single variable calculus. Here, the derivative of the function concerning \(x\) epitomizes how calculus aids in determining the rate of change.

• **Limits**: Establish how functions behave near certain points.
• **Derivatives**: Measure how functions change, central to finding rates of change.
• **Integrals**: Determine the area under curves, useful for cumulative measurements.

Each of these components is crucial for analyzing real-world scenarios where relationships between quantities need to be accurately modeled and understood. The study of single variable calculus thus opens doors to a thorough comprehension of such dynamics within and beyond mathematics.