\( \left. \begin{array} { l } { f ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { ( x + h ) - f ( x ) } { h } } \\ { f ( x ) = 5 x } \end{array} \right. \)

To find the derivative of the function \( f(x) = 5x \), we can plug this into the definition. Thus, \( f'(x) = \lim_{h \to 0} \frac{(x + h) - 5x}{h} = \lim_{h \to 0} \frac{h}{h} = \lim_{h \to 0} 1 = 1 \). So, the derivative \( f'(x) \) is constant at 5, meaning that the function has the same rate of change at any point. In the real world, derivatives are incredibly useful! For instance, if you were analyzing speed, the derivative of a position function gives you the velocity. In our case, since \( f(x) = 5x \) represents a linear function, its slope or rate of change is consistently 5—think of it as moving steadily at 5 units per time interval regardless of where you are on that line.