Mathematics

Problem

For all real numbers

x

in the domain of

\sec x

, the derivatives of

\sec x

exhibit a recurring structural pattern. (a)–(d) below guide you toward establishing a general statement valid for every positive integer

n

. a. Compute

\dfrac{d}{dx}\bigl(\sec x\bigr)

and

\dfrac{d^{2}}{dx^{2}}\bigl(\sec x\bigr)

, writing each derivative in factored form that isolates a single

\sec x

factor. b. Assume that for some positive integer

k

, the

k

th derivative of

\sec x

can be written in the form

\sec x\,P_k\bigl(\tan x\bigr)

, where

P_k

is a polynomial in

\tan x

with integer coefficients. Derive an explicit expression for the

(k+1)

st derivative in terms of

\sec x

and

\tan x

, and hence give an algebraic relation connecting

P_{k+1}

and

P_k

. (You need not determine the full polynomial.) c. Using your work in part (b), prove that for every positive integer

n

there exists a polynomial

P_n

with integer coefficients such that

\frac{d^{n}}{dx^{n}}\bigl(\sec x\bigr)=\sec x\,P_n\bigl(\tan x\bigr).

Clearly state the logical structure of your argument. d. State

P_3(t)

and

P_4(t)

explicitly by direct calculation, and verify that each satisfies the relation established in part (b).