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The power rule is a fundamental concept in calculus that simplifies the process of finding derivatives of functions. It's particularly useful when dealing with polynomials or functions which can be expressed in terms of powers of variables. The rule states that if you have a function in the form of \( x^n \), the derivative is given by \( nx^{n-1} \).
For example, if you start with \( f(x) = 5x^{\frac{4}{3}} \), applying the power rule involves taking the exponent \( \frac{4}{3} \), multiplying it by the coefficient \( 5 \), and then reducing the power by 1. Hence, the derivative becomes \( f'(x) = \frac{4}{3} \times 5x^{\frac{4}{3} - 1} = \frac{20}{3}x^{\frac{1}{3}} \).
This process not only makes differentiation easier but also allows quick computation which is essential for more complex calculus problems.

Derivative calculation is the process of finding the derivative of a function. This involves determining how the function changes as its input changes. In simple terms, it represents the rate of change or the slope of the function at any given point.
Using the power rule as illustrated, finding the derivative of \( f(x) = 5x^{\frac{4}{3}} \) requires applying the rule to yield \( f'(x) = \frac{20}{3}x^{\frac{1}{3}} \). After calculating the general derivative, the next step is to evaluate this derivative at a specific point. For this problem, we substitute \( x = 8 \) into the derived expression to find \( f'(8) \).
The computation \( f'(8) = \frac{20}{3} \times 8^{\frac{1}{3}} \) involves evaluating \( 8^{\frac{1}{3}} \) which is the cube root of 8, resulting in 2. So, \( f'(8) = \frac{20}{3} \times 2 = \frac{40}{3} \). This result represents the slope or rate of change of the function at the point where \( x = 8 \).

The domain of a function refers to all the possible input values (\( x \)-values) that allow the function to produce real number outputs. In most cases, unless specifically restricted, we assume the function's domain includes all real numbers where the function is defined.
For the function \( f(x) = 5x^{\frac{4}{3}} \), the expression inside the function needs to be real and non-negative for the cube root \( x^{\frac{1}{3}} \) to be valid. Fortunately, cube roots are defined for all real numbers, enabling the function to accept any real number value. Therefore, the domain of this function is all real numbers.
While working through derivatives, understanding the domain is vital for realistic interpretations, especially for ensuring that evaluated derivatives like \( f'(8) \) fall within the domain, thus confirming their applicability and correctness.