Real-Valued Derivatives
Derivatives tell us the rate of change of a function at any given point. They are extraordinarily useful because they allow us to do so for continuously changing functions (i.e. curves). In algebra, the slope formula only determines the average rate of change of a function between two intervals. In calculus, however, we can find the instantaneous rate of change at a point.
Let us try and understand the principle of a derivative with example.
Let us try and understand the principle of a derivative with example.
Let us obtain progressively accurate approximations of the rate of change of f(x) = -x(x-1)(x+1) at x = 1 by finding the average rate of change (which can be determined using an equation from algebra) between increasingly smaller intervals.
As we can see, as the interval gets increasingly smaller, the average slope of the function between that interval approaches the value -2. Thus, if we were to do this infinitely, the rate of change would be exactly 2. We can instead say that the limit as the change in x approaches 0 of f(x) is equal to -2.
Complex-Valued Derivatives
We can extend our definition of the derivative to the complex plane as well.
Rate of change can be envisioned as the difference quotient between the outputs and inputs.
Rate of change can be envisioned as the difference quotient between the outputs and inputs.