Unizor - Derivatives - Higher Order Derivatives

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):
lim_Δx→0[f(x₀+Δx)−f(x)]/Δx

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 derivativef ^I(x) is a subset of a domain of function f(x) since, in theory, the above limit might not exist at all points of domain of f(x). If this limit exists at each point of domain of f(x), the domain of a derivative f ^I(x) coincides with the domain of original function f(x).

At this point we can considerf ^I(x), the first derivative of function f(x), as a function in its own rights and differentiate it again, thus obtaining the second derivative of function f(x)denoted as f ^II(x).
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 f ^III(x) or d³f(x)/dx³, or d³/dx³[f(x)].

The next step is the fourth derivative of function f(x)denoted as f ^IV(x). And so on.

Let's illustrate this process with examples.

Example 1

f(x) = a (constant)

f ^I(x) = 0
f ^II(x) = 0
etc.

Example 2

f(x) = xⁿ

f ^I(x) = nxⁿ⁻¹
f ^II(x) = n(n−1)xⁿ⁻²
f ^III(x) = n(n−1)(n−2)xⁿ⁻³
f ^IV(x) = n(n−1)(n−2)(n−3)xⁿ⁻⁴
etc.

Example 3

f(x) = a^x

f ^I(x) = ln(a)·a^x
f ^II(x) = ln²(a)·a^x
f ^III(x) = ln³(a)·a^x
f ^IV(x) = ln⁴(a)·a^x
etc.

Example 4

f(x) = sin(x)

f ^I(x) = cos(x)
f ^II(x) = −sin(x)
f ^III(x) = −cos(x)
f ^IV(x) = sin(x)
etc.