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
dx
tan x = 1+tan2x
d2
dx2
tan x = 2tan x + 2tan3x
d3
dx3
tan x = 2+ 8tan2x + 6tan4x
d4
dx4
tan x = 16tan x + 40tan3x+24tan5x
d5
dx5
tan x = 16+ 136tan2x + 240tan4x+120tan6x

Observation:

(1) The successive derivatives of f(x) are polynomials of tan x. A closer examination suggests that the n-th order derivative of f(x) is of the form pn+1(tan x), where pn is a polynomial of degree n.

(2) Arranging the coefficients in the tabular each entry is seen as a sum of the upper left-hand and the upper right- hand entries mutiplied by the respective column numbers.

Reconstruction:

The coefficients of the polynomials pn 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
tann x = ntann-1x + ntann+1x,
from which the recurrence relation of the polynomial sequence {pn(z)}
p1(z) = z,     pn+1(z) = p¢n(z)(1+z2), 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. 254-255 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
dx
cotx = -1-cot2x
d2
dx2
cotx = 2cotx + 2cot3x
d3
dx3
cotx = -2- 8cot2x - 6cot4x
d4
dx4
cotx = 16cotx + 40cot3x+24cot5x
d5
dx5
cotx = -16- 136cot2x - 240cot4x-120cot6x

Explain the relationship between the two experiments.

(2) Plot the graphs of the polynomial sequence {pn(z)}