Simplify the expression for which the rate is desired. Perform any standard algebraic manipulations to simplify. Also, convert all radicals to fractional exponents. This allows for simple usage of the power rule throughout the process.
Determine the form of the final simplified function. Specifically, is it a product, sum, quotient, or nested series of terms. This determines which differentiation technique is required, the sum, product, quotient, or chain rule, respectively. Choosing an appropriate technique is a very heuristic process that may require some trial and error for even the simplest function.
Apply the appropriate differentiation technique to the original function.
Simplify the resulting function. This simplified version is known as the first derivative -- sometimes referred to as "f" prime of X -- of the function and is the rate of change (slope) of the function at any point along the domain of the original function.