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The power rule is a fundamental technique in calculus, immensely useful for finding the derivative of functions that can be expressed as a power of x.
Consider a function of the form f(x) = xⁿ, where n is any real number. The power rule says that the derivative of f with respect to x, denoted as f'(x), is nx^n-1.
For example, let’s differentiate f(x) = (x - 3)^2/3 using the power rule. Applying it directly, we get the derivative f'(x) = (2/3)(x - 3)^-1/3. This simplifies our differentiation process, making it a quick and powerful tool for handling polynomial functions and any other functions that can be written in the power form.

Understanding when a derivative is undefined is crucial in calculus, as it highlights the limitations of differentiability. A derivative may be undefined for several reasons:

• If the function has a sharp corner or cusp
• If the function has a vertical tangent at the point
• If the function is not continuous at the point

However, in the case of the function f(x) = (x - 3)^2/3', the only point of concern is the one where the expression inside the derivative becomes zero, as division by zero is undefined in mathematics. For the derivative f'(x) = (2/3)(x - 3)^-1/3, we can see that the expression (x - 3)^-1/3 will never be zero for real values of x. Hence, the derivative f'(x) does not become undefined for any real number except at x = 3, where the original function f(x) is not defined.

Calculus is deeply rooted in the set of real numbers. When we speak of real values in calculus, we're often interested in the behavior of functions within the real number system.
For continuous functions, like polynomials, we can usually find a derivative at every point where the function is defined. However, with more exotic functions, like our earlier example f(x) = (x - 3)^2/3, we may encounter complexities.
It's important to note that while f(x) is defined for all real values of x, the differentiability (the ability to find a derivative) of such functions can be restricted. In our example, the function is differentiable at all real numbers except at x = 3. This exclusion derives from the nature of the function, not the real number system. Understanding the behavior of functions over the real numbers enables us to grasp their properties, such as continuity and differentiability, which are key concepts in calculus.