The Derivatives of Higher Order of tan x

The first five sucessive derivatives of the function f(x) = tan x may be discovered by entering this single line in Derive:

From this the following pattern emerges:

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 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 left-hand 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

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

Explain the relationship between the two experiments.

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