QUESTION 7 (a) Use the method of first principles to determine the derivative of \( f(x)=\frac{6}{6} \)

The function \( f(x) = \frac{6}{6} = 1 \) is a constant function. To find the derivative using first principles, we consider the definition of the derivative: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Since \( f(x) \) is always 1, we can substitute: \[ f'(x) = \lim_{h \to 0} \frac{1 - 1}{h} = \lim_{h \to 0} \frac{0}{h} = 0 \] So, the derivative \( f'(x) = 0 \). In mathematics, constant functions are unique and make it easy to predict their behavior. They stay the same no matter what, and their derivatives illustrate this immutability, reinforcing the idea that the slope of a horizontal line is zero, as these functions graph as flat horizontal lines on the coordinate plane! Learning about derivatives not only benefits future calculus concepts but also unlocks various practical applications. In fields such as physics, economics, and engineering, the derivative represents how quantities change. For example, in physics, the derivative of position with respect to time gives you velocity—essentially how fast something is moving. Understanding this connection can lead to insights about motion, optimization, and change in numerous real-world scenarios!