Part 2/4:
Deriving the Quotient Rule
To derive the Quotient Rule, we start by defining ( u = f(x) ) and ( v = g(x) ). When we increment by ( \Delta x ), the new values become ( u + \Delta u ) and ( v + \Delta v ). The change in ( y ) can be expressed as the difference between these increments:
[
\Delta y = \frac{u + \Delta u}{v + \Delta v} - \frac{u}{v}.
]
To combine these fractions, we need a common denominator, which requires multiplying the terms appropriately. After careful simplification involving cancellation and rearranging, we arrive at an expression for ( \Delta y ) which can then be divided by ( \Delta x ) to find the average slope.