Derivative of Products & Quotients
Derivative of Product and Quotient The derivatives are widely used throughout Mathematics and the Sciences, and the Identities simplifying computations are invaluable tools. In this post, our intent is to derive the multiplication and division identities of derivatives using the very first principles. Derivative Suppose there be a continuous real function ` y = f(x) ` defined over an open interval `(a, b) \subseteq \mathbb{R}`. Then the derivative of the function `f(x)` at some point `c \in (a, b)` represented by, `f'(c) or \frac{d}{dx}f(x)` at `x = c` gives the rate of change of the function at the point `c` with respect to the independent variable. This idea however is still vague and we need a precise mathematical definition. Right-hand & Left-hand Rate: In the neighbour of `c`, let us define an interval `I = [c - h, c + h]` for some ` h > 0`. Then we may define the right-hand average rate of change of the function in the interval as, `\frac{f(c+h) - f(c)...