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Higher Derivatives
Suppose that there is the differentiable function y=f(x), i.e., there is the derivative y¢=f¢(x), which is denoted by f1(x). By Definition of the derivative we obtain
.
Definition. The derivative of the derivative of function is called the second derivative; the derivative of the second derivative of function is called the derivative of the third order respectively and is denoted by
,
,
.......
etc.
Notations:
, 
, etc., 
are read, for example, the third derivative – d three y with respect to the third power of dx.
If the order of derivative is noted in Arabic letters then it is written down in brackets.
In applications it is often required to find the higher derivatives of the product of two functions
(u.v)(n).
To derive this Leibniz formula, find the following derivatives of the first order
(u.v)¢=u¢v+uv¢,
and of the second order
(u.v)¢¢=u¢¢v+u¢v¢+u¢v¢+v¢¢u= u¢¢v+2u¢v¢+uv¢¢.
We differentiate once again and find the derivative of the third order:
(u.v)¢¢¢=u¢¢¢v+u¢¢v¢+2u¢¢v¢+2u¢v²+u¢v²+uv²¢= u¢¢¢v+3u¢¢v¢+3u¢v²+uv¢¢¢.
The derivative of the derivative of two functions is said to be the Binomial theorem if the inverse order is taken nominally as the power of this function:
(u.v)(n)=u(n). v+Cn1. u(n-1). v¢+ Cn2. u(n-2). v¢¢+... +
+ Cnn-1.u¢v(n-1)+Cnn.uv(n).
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