*Notes to a video lecture on http://www.unizor.com*

__Higher Order Derivatives__

We have introduced a concept of

*derivative*at point

**of a real function**

*x*_{0}**defined in some**

*f(x)**neighborhood*of this point as the following limit (if it exists):

*lim*_{Δx→0}

**[**Δ

*f(x*_{0}+**Δ**

*x)−f(x)*]*/*

*x*The limit defining a derivative of function

**at point**

*f(x)***, if it exists, is some real value defined by both the function itself and a point**

*x*_{0}**in its domain.**

*x*_{0}Therefore, for all these points of a domain of function

**where this limit exists the derivative is a new function defined at all these points.**

*f(x)*This new function, a derivative of function

**, defined at all points of a domain of function**

*f(x)***where the above limit exists, is traditionally denoted as**

*f(x)*

*f**and called the*

^{ I}**(x)***first derivative*of function

**.**

*f(x)*The domain of the derivative

*f*

^{ I}**(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(x)*

*f**coincides with the domain of original function*

^{ I}**(x)****.**

*f(x)*At this point we can consider

*f*

^{ I}**(x)****, as a function in its own rights and differentiate it again, thus obtaining the**

*f(x)**second derivative*of function

**denoted as**

*f(x)*

*f*

^{ II}**(x)**Alternative notation for the second derivative is

*d²*or

**f(x)/**dx²*d²/dx²*.

**(f(x))**This process of derivation can be continued resulting in the

*third derivative*of function

**denoted as**

*f(x)*

*f*

^{ III}**(x)***d³*, or

**f(x)/**dx³*d³/dx³*.

**[f(x)]**The next step is the

*fourth derivative*of function

**denoted as**

*f(x)*

*f*

^{ IV}**(x)**Let's illustrate this process with examples.

*Example 1*

**(constant)**

*f(x) = a*

*f*

^{ I}**(x) = 0**

*f*

^{ II}**(x) = 0**etc.

*Example 2*

*f(x) = x*^{n}

*f*

^{ I}**(x) = nx**^{n−1}

*f*

^{ II}**(x) = n(n−1)x**^{n−2}

*f*

^{ III}**(x) = n(n−1)(n−2)x**^{n−3}

*f*

^{ IV}**(x) = n(n−1)(n−2)(n−3)x**^{n−4}etc.

*Example 3*

*f(x) = a*^{x}

*f*

^{ I}**(x) = ln(a)·a**^{x}

*f*

^{ II}**(x) = ln**^{2}(a)·a^{x}

*f*

^{ III}**(x) = ln**^{3}(a)·a^{x}

*f*

^{ IV}**(x) = ln**^{4}(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.

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