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The Power Rule is one of the most fundamental tools in calculus differentiation. It provides a simple way to find the derivative of polynomial functions, which are functions consisting of terms like powers of variables. The Power Rule states that if you have a function of the form \(x^n\), where \(n\) is any real number, its derivative is \(nx^{n-1}\). This makes it particularly easy to differentiate terms like \(20x^3\) and \(-36x^2\).

• For \(20x^3\), applying the Power Rule means multiplying the coefficient (20) by the exponent (3), and then reducing the exponent by 1, resulting in \(60x^2\).
• For \(-36x^2\), multiply \(-36\) by the exponent (2), and reduce the exponent by 1, yielding \(-72x\).

In essence, the Power Rule is a quick way to determine how the function changes, or its rate of change at any given point. It simplifies the process of finding higher-order derivatives like the second or third, by allowing you to apply the rule repeatedly.

Differentiation in calculus is a process used to find the derivative of a function. A derivative represents the rate at which a quantity changes. For example, in physics, derivatives can determine the velocity of an object from its position function. In our exercise, finding the higher-order derivative involves multiple rounds of differentiation. It's key to:

• Identify which function or derivative you are differentiating. Here, we start from the second derivative \(f''(x)\) and find the third derivative \(f'''(x)\).
• Apply the differentiation rules correctly, such as the Power Rule.
• Combine the differentiated terms properly.

Calculus differentiation allows us to analyze the behavior of functions, find slopes of tangents, and solve real-world problems related to rates of change and motion.

Polynomial functions are expressions consisting of variables raised to whole number powers and coefficients. They appear frequently in mathematics because they are simple yet powerful models of relationships between variables. These functions have the form:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\]In our exercise, we dealt with a second derivative function \(f''(x) = 20x^{3} - 36x^{2}\). This represents a polynomial function because it involves powers of \(x\) with real-number coefficients.

• Polynomials are easy to differentiate using rules like the Power Rule.
• Each term is differentiated separately, then combined to form a new polynomial function.

Polynomial functions and their derivatives have key applications in engineering, economics, and physics due to their ability to model smooth curves and predict trends in data.