THE FORMULA FOR HIGHER ORDER DERIVATIVES OF INVERSE FUNCTIONS

1 THE FORMULA FOR HIGHER ORDER DERIVATIVES OF INVERSE FUNCTIONS

THE FORMULA FOR HIGHER ORDER DERIVATIVES OF INVERSE FUNCTIONS

By Ryuji Kaneiwa

Theorem. Let n be a positive integer. If y is a C ⁿ -function of x and ^dy _dx 0 in a certain interval, then

d ⁿ x dy ⁿ

1 ^{n 1}

dy dx 2n 1

s1 s2 n 1 1s1 2s2 2n 2

1 ^s

2n s 1 2 ! ^dy _dx ^s

^d _dx

^y

s

2! ^s

s 2 ! 3! ^s

s 3 ! is valid in the same interval.

proof. Set that

(1) f y, x h x y

and let x g y be the inverse function of y h x . Then x g y is the implicit function determined by f y, x 0. By the formula in the note[1],

(2) g ⁿ y 1

f x 2n 1

u P

n,2n 2

ν u f u; y, x , where N 0 0, 1, 2, . . . , N 2 N 0 2 0, 0 ,

P l n 1 , n 2

u N 0 N

;

i,j N

u i, j l,

i,j N

i u i, j n 1 ,

i,j N

j u i, j n 2 , ν u 1 ^{u 0,1} u 0, 1 ! 2n u 0, 1 2 ! n!

i,j N

i! j! ^{u i,j} u i, j ! and

f u; y, x

i,j N

f ^i,j y, x ^{u i,j} .

f

y, x

f y, x

THE FORMULA FOR HIGHER ORDER DERIVATIVES OF INVERSE FUNCTIONS

By

Ryuji KANEIWA

2

Let K be a set such that

K i, j N 2 ; i 1 or i 1, j 0 . From (1), we have for i, j N 2 ,

(3) f ^i,j y, x

h ^j x , if i 0, j 0 1 , if i, j 1, 0 , 0 , if i, j K.

If i, j K and u i, j 0, then f u; y, x 0.

Therefore,

g ⁿ y 1

f x 2n 1

u P2n 1n,2n 2 i,j K u i,j 0

ν u f u; y, x .

For the proof of the theorem, we needs following two lemmata.

Lemma 1. If u P _{2n 1} n, 2n 2 and u i, j 0 for i, j K , then u 1, 0 n,

j 1

u 0, j n 1 and

j 1

j u 0, j 2n 2.

Lemma 2. If a sequence s j j N of non-negative integers satisfying s j n 1 and js j 2n 2, then the map u N 0 N

defined by (4) u i, j

s j , if i 0, j 0 n , if i, j 1, 0 , 0 , if i, j K is an element of P _{2n 1} n, 2n 2 .

We can easily get these lemmata. By Lemma 1 and Lemma 2,

g ⁿ y 1

f x 2n 1

s1 s2 n 1 1s1 2s2 2n 2

ν u f u; y, x .

with u defined by (4). Thus we have

3 THE FORMULA FOR HIGHER ORDER DERIVATIVES OF INVERSE FUNCTIONS

ν u 1 ^s

s 1 ! 2n s 1 2 ! n!

1! 0! ⁿ n! 1! ^s

s 1 ! 2! ^s

s 2 !

1 ^s

2n s 1 2 ! 2! ^s

s 2 ! 3! ^s

s 3 ! and

f u; y, x f ^1,0 y, x ⁿ f ^0,1 y, x ^s

f ^0,2 y, x ^s

1 ⁿ h x ^s

h x ^s

This completes the proof.

Reference

[1] Kaneiwa, R., The Formula of Higher Order Derivatives of Implicit Functions, The Review of Liberal Arts, No.129, 1-19 (2015), Otaru Uni- versity of Commerce.

Mogami 2-21-10

Otaru, Hokkaido

047-0023 Japan