\( [ f ^ { \prime } ( x ) = \frac { ( x - 7 ) ( x + 2 ) ^ { 2 } } { ( x - 1 ) ^ { 3 } } ] \)

Fun fact: The expression you're dealing with is the derivative of a function, which often indicates important information about the function’s behavior, such as where it increases, decreases, or has critical points. In this case, the zeros of the numerator (x - 7) and (x + 2)^2 will guide you to the x-values where the slope changes—like a rollercoaster revealing its peaks and valleys! When solving this derivative, it's easy to overlook signs and critical points. A common mistake is not checking the intervals around those critical values, leading to incorrect conclusions about the function's trends. Always remember to consider the sign of the derivative on intervals: positive means the function is increasing and negative indicates it's decreasing. It's like mapping out a treasure hunt—seeking out those clues!