Editing Derivative (section)

== Higher-order derivatives <span class="anchor" id="order of derivation"></span><span class="anchor" id="Order"></span> ==
''Higher order derivatives'' are the result of differentiating a function repeatedly. Given that <math> f </math> is a differentiable function, the derivative of <math> f </math> is the first derivative, denoted as {{tmath|1= f' }}. The derivative of <math> f' </math> is the [[second derivative]], denoted as {{tmath|1= f' ' }}, and the derivative of <math> f'' </math> is the [[third derivative]], denoted as {{tmath|1= f' ' ' }}. By continuing this process, if it exists, the {{tmath|1= n }}th derivative is the derivative of the {{tmath|1= (n - 1) }}th derivative or the ''derivative of order {{tmath|1= n }}''. As has been [[#Notation|discussed above]], the generalization of derivative of a function <math> f </math> may be denoted as {{tmath|1= f^{(n)} }}.{{sfnm
 | 1a1 = Apostol | 1y = 1967 | 1p = 160
 | 2a1 = Varberg | 2a2 = Purcell | 2a3 = Rigdon | 2y = 2007 | 2pp = 125–126
}}  A function that has <math> k </math> successive derivatives is called ''<math> k </math> times differentiable''. If the {{nowrap|1=<math> k </math>-}}th derivative is continuous, then the function is said to be of [[differentiability class]] {{tmath|1= C^k }}.{{sfn|Warner|1983|p=5}} A function that has infinitely many derivatives is called ''infinitely differentiable'' or ''[[smoothness|smooth]]''.{{sfn|Debnath|Shah|2015|p=[https://books.google.com/books?id=qPuWBQAAQBAJ&pg=PA40 40]}} Any [[polynomial]] function is infinitely differentiable; taking derivatives repeatedly will eventually result in a [[constant function]], and all subsequent derivatives of that function are zero.{{sfn|Carothers|2000|p=[https://books.google.com/books?id=4VFDVy1NFiAC&pg=PA176 176]}}

One [[Differential calculus#Applications of derivatives|application of higher-order derivatives]] is in [[physics]]. Suppose that a function represents the position of an object at the time. The first derivative of that function is the [[velocity]] of an object with respect to time, the second derivative of the function is the [[acceleration]] of an object with respect to time,{{sfn|Apostol|1967|p=160}} and the third derivative is the [[jerk (physics)|jerk]].{{sfn|Stewart|2002|p=193}}