Notes to a video lecture on http://www.unizor.com
Higher Order Derivatives
We have introduced a concept of derivative at point x0 of a real function f(x) defined in some neighborhood of this point as the following limit (if it exists):
The limit defining a derivative of function f(x) at point x0, if it exists, is some real value defined by both the function itself and a point x0 in its domain.
Therefore, for all these points of a domain of function f(x)where this limit exists the derivative is a new function defined at all these points.
This new function, a derivative of function f(x), defined at all points of a domain of functionf(x) where the above limit exists, is traditionally denoted as f I(x) and called the first derivative of function f(x).
The domain of the derivative
At this point we can consider
Alternative notation for the second derivative is d²f(x)/dx²or d²/dx²(f(x)).
This process of derivation can be continued resulting in thethird derivative of function f(x)denoted as
The next step is the fourth derivative of function f(x)denoted as
Let's illustrate this process with examples.
f(x) = a (constant)
f(x) = xn
f(x) = ax
f(x) = sin(x)