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Differential Calculus

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5 Sep 2007 16:30:37 IST

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WHAT IS NTH DERIVATIVE OF TAN-X

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WHAT IS NTH DERIVATIVE OF TAN-X

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venkat v.t

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6 Sep 2007 17:41:44 IST

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Let y=tanx

y=sinx/cosx then cross multiply and differentiate n times
both sides.

ninad pethkar

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6 Sep 2007 19:06:27 IST

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sec's sqare x

venkat v.t

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6 Sep 2007 21:45:42 IST

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Hello Ninz.First read the question properly.

Avinash Sharma

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12 Sep 2007 03:38:48 IST

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The Derivatives of Higher Order of tan x

Discovery Process:

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

d tan x/ dx = 1+tan²x

d² tan x/dx² = 2tan x + 2tan³x

d³ tan x/dx³ = 2+ 8tan²x + 6tan⁴x

d⁴ tan x/dx⁴ = 16tan x + 40tan³x+24tan⁵x

d⁵ tan x/dx⁵ = 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 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 multiplied 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

dⁿ tan x/dxⁿ = ntan^n-1x + ntanⁿ⁺¹x,

from which the recurrence relation of the polynomial sequence {p_n(z)}

p₁(z) = z, p_n+1(z) = p'_n(z)(1+z²), n ³ 1 is then formulated.

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