QUESTION ONE ( 16 MARKS) (a) Compute the derivative of \( y=\frac{1}{x} \) using the first principle (or by definition).

To compute the derivative of \( y = \frac{1}{x} \) using the first principle, we start with the definition of the derivative: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] Here, \( f(x) = \frac{1}{x} \). Therefore, \[ f(x+h) = \frac{1}{x+h} \] Now, substituting into our derivative formula: \[ f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \] Finding a common denominator for the terms in the numerator: \[ f'(x) = \lim_{h \to 0} \frac{\frac{x - (x+h)}{(x+h)x}}{h} = \lim_{h \to 0} \frac{\frac{-h}{(x+h)x}}{h} \] Simplifying further: \[ = \lim_{h \to 0} \frac{-1}{(x+h)x} \] As \( h \) approaches 0, we have: \[ = \frac{-1}{x^2} \] Thus, the derivative is: \[ f'(x) = -\frac{1}{x^2} \] Now let's dive into some interesting facts! Throughout history, calculus was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. This groundbreaking work laid the foundations for modern mathematics and science. Their ideas allowed for the rigorous treatment of rates of change, integrating concepts that would become crucial for developments in physics, engineering, and economics. Imagine a world without derivatives to analyze motion or change—it would likely look very different! If you're applying the concept of derivatives in real-world contexts, think about physics and engineering! For example, engineers use derivatives to find the maximum and minimum points of structures, ensuring stability and efficiency. They apply calculus to optimize materials and designs, translating mathematical knowledge into practical inventions like bridges and airplanes. Understanding how to calculate derivatives opens up a world of possibilities in your career path!