The Derivatives of
Higher Order of tan x
Discovery Process:
The first five sucessive derivatives of the function f(x) = tan x may
be discovered by entering this single line in Derive:
VECTOR(DIF(TAN(x),x,k),k,1,5)
From this the following pattern emerges:

d^{2
}dx^{2} 
tan x = 2tan x + 2tan^{3}x 


d^{3
}dx^{3} 
tan x = 2+ 8tan^{2}x + 6tan^{4}x 


d^{4
}dx^{4} 
tan x = 16tan x + 40tan^{3}x+24tan^{5}x 


d^{5
}dx^{5} 
tan x = 16+ 136tan^{2}x + 240tan^{4}x+120tan^{6}x 

Observation:
(1) The successive derivatives of f(x) are polynomials of tan x. A closer
examination suggests that the nth order derivative of f(x) is of the form
p_{n+1}(tan x), where p_{n} is a polynomial of degree n.
(2) Arranging the coefficients in the tabular each entry is seen as
a sum of the upper lefthand and the upper right hand entries mutiplied
by the respective column numbers.
Reconstruction:
The coefficients of the polynomials p_{n} may be reconstructed
by means of a spreadsheet. After the entry 1 indicating the obvious identity
tan x = tan x is placed, there is only one formula to be entered and then
copied: The reconstruction is based on the formula

d
dx 
tan^{n} x = ntan^{n1}x + ntan^{n+1}x, 

from which the recurrence relation of the polynomial sequence {p_{n}(z)}
p_{1}(z) = z, p_{n+1}(z)
= p^{¢}_{n}(z)(1+z^{2}),
n ³ 1 

is then formulated.
Related Topics in the Calculus Course: derivatives of higher
order, Leibniz's formula for the nth derivative of a product.
Software Used: Derive or Maple may be used for discovering the
pattern in the successive derivatives, the spreadsheet may be used for
tabulation of coefficients.
Reference: The result is to be compared with the discussion on
pp. 254255 of J. Pierpont, The
Theory of Function of Real Variables, Volume 1.
Further Investigations:
(1) Changing the function f(x) from tan x to cot x and repeat the same
procedure as above, the resulting table of integration becomes

d^{2
}dx^{2} 
cotx = 2cotx + 2cot^{3}x 


d^{3
}dx^{3} 
cotx = 2 8cot^{2}x  6cot^{4}x 


d^{4
}dx^{4} 
cotx = 16cotx + 40cot^{3}x+24cot^{5}x 


d^{5
}dx^{5} 
cotx = 16 136cot^{2}x  240cot^{4}x120cot^{6}x 

Explain the relationship between the two experiments.
(2) Plot the graphs of the polynomial sequence {p_{n}(z)}