HIGHER ORDER DERIVATIVES OF FUNCTIONS DEPENDING ON OTHER FUNCTIONS

DEPENDING ON OTHER FUNCTIONS

ION CHIT¸ ESCU

For functions f : I → R, g : I → Ror C, f injective, one computes the n-th derivative ofF :f(I) →R or C, F(x) = g(f⁻¹(x)). To this end, one uses the formula of Fa`a di Bruno and a formula for higher order derivatives of inverses by the present author. Three different formulae are given.

AMS 2010 Subject Classification: 26A04, 26A24.

Key words: generalized composition, generalized inverse, higher order derivatives.

1. INTRODUCTION

For an intervalI and functionsf :I →R,g:I →C, the injectivity off implies functional dependence, i.e. one can consider the functionF :f(I)→C, F(x) =g(f⁻¹(x)), the inverse of f being defined on the image f(I).

Alternatively, writing, fort∈I:

x=f(t), y=g(t) one can eliminate t, obtaining y=F(x).

The aim of this paper is to give formulae for then-th derivativeDⁿF(x) in terms of the derivatives D^pf(t) and D^pg(t), p ≤ n. In case g(t) = t, one obtains formulae for the n-th derivative off⁻¹ in terms of D^pf.

Besides the purely theoretical interest, the solution of this problem can be useful for the study of parametrical representations of the form x =f(t), y =g(t), in order to draw the representative images and to put into evidence the underlying properties.

The present paper relies heavily on our previous paper [4] and can be viewed as a continuation of it. The major result used in both papers is the fa- mous formula of Francesco Fa`a di Bruno exibiting then-th derivativeDⁿ(g◦f) in terms of D^pf and D^pg.

Using this formula and the formula for Dⁿ(f⁻¹) from [4] we give three formulae for DⁿF.

The first formula is the major result of the paper. The second formula is a more complicated version of the first formula. The third formula is, as a

REV. ROUMAINE MATH. PURES APPL.,57(2012),2, 105-115

matter of fact, an inductive method, not using the formula of Fa`a di Bruno.

It uses polynomials in several variables and can be easily applied.

2. PRELIMINARY FACTS

Throughout the paper, Nwill be the set of natural numbers with N^∗ = N\ {0},K =R orCwill be the set of scalars (real or complex) andI,J will be (non degenerate) intervals of real numbers.

For non empty setsX, Y,Z,A⊂X, B ⊂Y and functions f :X →Y such thatf(A)⊂B,g:B →Z, one can construct the generalized composition g◦f :A→Z, given via (g◦f)(x) =g(f(x)).

For non empty sets X, Y and injective f : X → Y, one can construct the generalized inverse f⁻¹ of f which is the function h :f(X) → X, given via h(y) =x, wherex∈X is uniquely determined by the conditionf(x) =y.

Let 1≤k≤nbe natural numbers. An element (k₁, k₂, . . . , k_n)∈N will be called (n, k)-multiindex if it has the following properties:

k₁+k₂+· · ·+k_n=k, k₁+ 2k₂+ 3k₃+· · ·+nk_n=n.

We shall write

M(n, k) = the set of all (n, k)-multiindexes.

For instance

M(1,1) ={(1)}, M(2,1) ={(0,1)}, M(2,2) ={(2,0)},

M(4,2) ={(1,0,1,0),(0,2,0,0)}.

For any 1≤k≤n, the setM(n, k) is non empty. The (n, k)-multiindexes have combinatorial interpretation.

For p ∈ N^∗, the p-derivative of a function u at x will be denoted by D^pu(x). Incidentally we shall also write

D¹u(x) =f⁰(x), D²u(x) =f⁰⁰(x), D³u(x) =f⁰⁰⁰(x).

Now, we are ready to present the famous formula of Fa`a di Bruno (ori- ginally in [2] and [3]). See also [4], [5] and [7].

Theorem 1(The formula of Fa`a di Bruno). Let I, J be intervals, t∈I and n∈N^∗. Let f :I → R be n times differentiable at t, such that f(I) = J and let g:J →K be n times differentiable at f(t). Then g◦f :I →K isn

times differentiable at tand one has Dⁿ(g◦f)(f) =

n

X

k=1

D^kg(f(t))·A(n, k)(t), where

A(n, k)(t) = X

(k1,k2,...,kn)∈M(n,k)

n!

k1!k2!. . . kn!

D¹f(t) 1!

k1

·

D²f(t) 2!

k2

·. . .·

Dⁿf(t) n!

kn

.

In [4], we proved

Theorem 2 (The Formula for Higher Order Derivatives of Inverses).

Let I be an interval, f : I → R be continuous and strictly monotone and let J = f(I). Let t ∈ I be such that f is n times differentiable at t, n ≥ 2 and f⁰(t)6= 0.

Thenh=f⁻¹ :J →I is ntimes differentiable at x=f(t) and one has, for any 1≤m≤n

D^mh(x) = (−1)^m+1 D_m(t) (f⁰(t))^m(m+1)2

. Here (see Theorem 1)

D₁(t) = 1,

D₂(t) =A(2,1)(t), D₃(t) =

A(2,1)(t) A(2,2)(t) A(3,1)(t) A(3,2)(t)

, and for m≥4

Dm(t) =

A(2,1)(t) A(2,2)(t) 0 0 . . . . . .0

A(3,1)(t) A(3,2)(t) A(3,3)(t) 0 . . . . . .0 . . . .

A(m−1,1)(t) A(m−1,2)(t) A(m−1,3)(t) . . . A(m−1, m−1)(t)

A(m,1)(t) A(m,2)(t) A(m,3)(t) . . . A(m, m−1)(t) .

3. RESULTS

Troughout this paragraph, we shall work within the following

Framework. AssumeI ⊂Ris an interval, f :I →R is continuous and strictly monotone and g : I → K. One can consider the generalized inverse h=f⁻¹ :J =f(I)→I and the functionF =g◦h:J →R.

We write, fort∈I:

f(t) =x, g(t) =y,

hence, for x ∈ f(I), we can write h(x) = t and F(x) = y (and even worse:

F(x) =y(x)).

To complete the framework, we shall consider n ∈ N^∗ and to ∈ I such that f and g arentimes differentiable att₀ and f⁰(t₀)6= 0.

Accepting this framework, we shall give formulae for DⁿF(x₀), where x₀=f(t₀).

We shall accept the well-known fact that, within the above mentioned framework, the function h isntimes differentiable atx₀.

Now we are in position to prove (induction onn)

Theorem 3. The function F isn times differentiable at x₀. For the first derivatives, one has

F⁰(x₀) = g⁰(t₀) f⁰(t0), (1)

F⁰⁰(x0) = g⁰⁰(t0)f⁰(t0)−g⁰(t0)f⁰⁰(t0) f⁰(t0)³ . (2)

Proof. The result is valid forn= 1. Indeed, let (x_n)_n⊂f(I)\ {x₀}be a sequence such that xn−→

n x0, wherexn=f(tn). Because h is continuous, one has t_n−→

n t₀, hence

F(x_n)−F(x₀)

x_n−x₀ = g(t_n)−g(t₀) f(t_n)−f(t₀) −→

n

g⁰(t₀) f⁰(t₀) proving (1).

Accept the result for n and let us prove it for n+ 1. Because f and g are n+ 1 times differentiable at t0, they are n times differentiable in a neighbourhood U of t₀ and f⁰(t) 6= 0 in U, because f⁰ is continuous at t₀. Using stepn= 1,F is differentiable in the neighbourhoodf(U) ofx₀ and, for any x∈f(U), one has (write x=f(t),t∈U)

(3) F⁰(x) = g⁰(t)

f⁰(t) = g⁰(h(x)) f⁰(h(x)) =

g⁰ f⁰ ◦h

(x) (generalized composition).

Using (3) and the fact thathisntimes differentiable inf(U) (according to the fact that f is n times differentiable in U and f⁰(t) 6= 0 for t∈ U), we see that F⁰ isn times differentiable atx0.

Equality (2) is obtained using (3).

First Formula for DⁿF(x₀)

We have F(x) = g(h(x)) in a neighbourhood of x₀. Using the Formula of Fa`a di Bruno, we get (forn∈N^∗)

DⁿF(x0) =

n

X

k=1

D^kg(h(x0))·b(n, k)(x0), where

b(n, k)(x0) = X

(k1,k2,...,kn)∈M(n,k)

n!

k₁!k₂!. . . k_n!·

·

D¹h(x0) 1!

k1

D²h(x0) 2!

k2

·. . .·

Dⁿh(x0) n!

kn

. Because (for 1≤p≤n) one has (see Theorem 2)

D^ph(x₀) = (−1)^p+1 Dp(t0) f⁰(t₀)^p(p+1)2 we get

DⁿF(x0) =n!

n

X

k=1

D^kg(t0)B(n, k)(t0), where

B(n, k)(t0) = X

(k1,k2,...,kn)∈M(n,k)

1

k1!. . . kn!·(−1)^(1+1)k1

(1!)^k1 · D₁(t₀) f⁰(t0)^1·22

!k1

·

·(−1)^(2+1)k2

(2!)^k2 · D₂(t₀) f⁰(t0)^2·32

!k2

·. . .·(−1)^(n+1)kn

(n!)^kn · D_n(t₀) f⁰(t0)^n(n+1)2

!kn

. Because

(1 + 1)k1+ (2 + 1)k2+· · ·+ (n+ 1)kn=n+k we obtain finally our

FIRST FORMULA

(∗) DⁿF(x₀) =n!(−1)ⁿ

n

X

k=1

D^kg(t₀)C(n, k)(t₀), where

C(n, k)(t₀) = (−1)^k X

(k1,k2,...,kn)∈M(n,k)

1

k1!(1!)^k1 ·k2!(2!)^k2 ·. . .·kn!(n!)^kn·

· D₁(t₀) f⁰(t₀)^1·22

!k1

· D₂(t₀) f⁰(t₀)^2·32

!k2

·. . .· D_n(t₀) f⁰(t0)^n(n+1)2

!kn

.

PARTICULAR CASE

When g(t) = t for all t in I, one gets F(x) = h(x) and (∗) becomes (because D^kg(t)6= 0⇒k= 1 and M(n,1) ={(0,0, . . . ,0,1)})

Dⁿh(x0) =n!(−1)ⁿ·(−1)¹· 1

n!· Dn(t0) f⁰(t₀)^n(n+1)2

!1

= (−1)ⁿ⁺¹ Dn(t0) f⁰(t₀)^n(n+1)2 confirming the previous formula.

Second Formula for DⁿF(x₀)

A.We start by considering the same intervalI and the pointt₀ ∈I. Let u : I → R, v :I → Rbe two functions such that v(t) 6= 0 for any t ∈I and assume u and v areq times differentiable att₀,q ∈N^∗.

We shall compute D^{q u}_v (t₀).

Using the formula of Leibniz we obtain

(4) D^q

u v

(t0) =

q

X

p=0

q!

p!(q−p)!D^p 1

v

(t0)·D^q−p(u)(t0).

We have

1

v =ϕ◦v, where

ϕ:R\ {0} →R, ϕ(x) = 1 x and, for any k∈N^∗

D^kϕ(x) = (−1)^k·k!·x^−k−1.

Hence, using Fa`a di Bruno’s Formula, we get for any 1≤p≤q D^p

1 v

(t₀) =

p

X

k=1

(−1)^k·k!·v(t)^−k−1· (5)

· X

(k1,k2,...,kp)∈M(p,k)

p!

k1!·k2!·. . .·kp!·

D¹v(t) 1!

k1

·. . .·

D^pv(t) p!

kp

and (4) gives

(6) D^qu

v

(t₀) = q!

0!q!· 1

v(t₀)D^qu(t₀)+

+

q

X

p=1

q!

p!(q−p)!D^p 1

v

(t0)·D^q−p(u)(t0) =

=q!

D^qu(t0) q!

1 v(t₀)+

q

X

p=1

D^q−pu(t0) (q−p)! · 1

p!D^p 1

v

(t₀)

. Using(5) and (6), we get

(7) D^q u

v

(t0) =q!

D^qu(t0) q! · 1

v(t₀) +

q

X

p=1

D^q−pu(t0)

(q−p)! ·Wp(t0)

, where

W_p(t₀) =

p

X

k=1

(−1)^k·k!· 1 v(t)^k+1·

· X

(k1,k2,...,kp)∈M(p,k)

1

k1!·k2!·. . .·kp!·

D¹v(t₀) 1!

k1

·. . .·

D^pv(t₀) p!

kp

. We think formula (7) can be of independent interest.

B. Let us come back to the initial framework. We have seen (formula (3)) that, for x in a neighourhood ofx₀, one has

D¹F(x) =F⁰(x) = g⁰

f⁰ ◦h

(x) hence (again Fa`a di Bruno’s Formula)

DⁿF(x₀) =Dⁿ⁻¹(D¹F)(x₀) =Dⁿ⁻¹ g⁰

f⁰ ◦h

(x₀) = (8)

=

n−1

X

q=1

D^q g⁰

f⁰

(h(x₀) =t₀) ·b(n−1, q)(x₀),

where (practically, we repeat the notation) b(n−1, q)(x₀) =

= X

(a1,...,an−1)∈M(n−1,q)

(n−1)!

a1!·. . .·an−1!·

D¹h(x₀) 1!

a1

·. . .·

Dⁿ⁻¹h(x₀) (n−1)!

an−1

. We lay stress upon the fact that formula (8) is valid forn≥2.

In order to write explicitely (8), we use (7) with u = g⁰, v = f⁰, the formula

D^mh(x₀) = (−1)^m+1 D_m(t₀) f⁰(t0)^m(m+1)2

and the fact that, for (a₁, a₂, . . . , an−1)∈M(n−1, q) one has 2a₁+ 3a₂+· · ·+nan−1 =

=a₁+ 2a₂+· · ·+ (n−1)an−1+a₁+a₂+· · ·+an−1 = (n−1) +q.

We get, finally, our

SECOND FORMULA (valid forn≥2)

(∗∗) DⁿF(x₀) = (n−1)!(−1)ⁿ⁻¹

n−1

X

q=1

q!(−1)^qA_q(t₀), where

A_q(t₀) =B_q(t₀) X

(a1,...,an−1)∈M(n−1,q)

1

a1!(1!)^a1 ·. . .·an−1!((n−1)!)^an−1·

· D₁(t₀) f⁰(t0)^1·22

!a1

·. . .· Dn−1(t₀) f⁰(t0)^(n−1)n2

!an−1

and

B_q(t₀) = D^q+1g(t0)

q! · 1

f⁰(t₀) +

q

X

p=1

D^q−p+1g(t0) (q−p)! ·

p

X

k=1

(−1)^k·k! 1 f⁰(t₀)^k+1 ·

·





X

(k1,...,kp)∈M(p,k)

1 k1!·. . .·kp!

D²f(t₀) 1!

k1

·. . .·

D^p+1f(t₀) p!

kp







. Comment.Of course, formula (∗∗) is much more complicated than for- mula (∗) and, from practical point of view, less useful. We think this formula can be used to obtain some new identities.

Third Formula for DⁿF(x0)

In order to present our third formula, we introduce the sequence (Pn)n≥1

of uniquely determined real polynomialsPn(X1, X2, . . . , Xn;Y1, Y2, . . . , Yn) in 2nvariables, given as follows:

(i) P₁(X₁;Y₁) =Y₁,

Pn+1(X1, X2, . . . , Xn+1;Y1, Y2, . . . , Yn+1) = (ii)

=

n

X

i=1

∂Pn(X1, X2, . . . , Xn;Y1, Y2, . . . , Yn)

∂X_i ·X1Xi+1+

+

n

X

j=1

∂Pn(X1, X2, . . . , Xn;Y1, Y2, . . . , Yn)

∂Y_j ·X1Yj+1−

−(2n−1)P_n(X₁, X₂, . . . , X_n;Y₁, Y₂, . . . , Y_n)·X₂. The first three polynomials:

P₁(X₁;Y₁) =Y₁⇒ ∂P₁(X₁;Y₁)

∂X1

= 0 and ∂P₁(X₁;Y₁)

∂Y1

= 1⇒

⇒P2(X1, X2;Y1, Y2) =X1Y2−X2Y1

P₃(X₁, X₂, X₃;Y₁, Y₂, Y₃) =X₁²Y₃+ 3X₂²Y₁−X₁X₃Y₁−3X₁X₂Y₂. THIRD FORMULA

DⁿF(x₀) = (∗ ∗ ∗)

= Pn(D¹f(t0), D²f(t0), . . . , Dⁿf(t0);D¹g(t0), D²g(t0), . . . , Dⁿg(t0))

(D¹f(t₀))²ⁿ⁻¹ .

The proof will be performed via induction onn.

Forn= 1, our assertion is true (see (1)):

D¹F(x0) = D¹g(t0) D¹f(t₀).

Let us accept the assertion for n and let us prove it for n+ 1. So, our hypothesis is thatfandgaren+1 times differentiable att₀, withD¹f(t₀)6= 0.

It follows that f and g aren times differentiable in a neighbourhoodU of t0, with D¹f(t)6= 0 for allt∈U (continuity ofD¹f).

The induction hypothesis exhibits a n-degree homogeneous polynomial Pn(X1, X2, . . . , Xn;Y1, Y2, . . . , Yn) such that, for anyx=f(t)∈f(U), one has

DⁿF(x) = P_n(A_n(t)) f⁰(t)²ⁿ⁻¹ ,

where An(t) = (D¹f(t), D²f(t), . . . , Dⁿf(t);D¹g(t), D²g(t), . . . , Dⁿg(t)).

Writingu:U →K,u(t) =P_n(A_n(t)), we get DⁿF(x) = u(t)

f⁰(t)²ⁿ⁻¹. It follows (becauseD¹h(x0) = (f⁰(t0))⁻¹) that

Dⁿ⁺¹F(x₀) =D¹

u (f⁰)²ⁿ⁻¹

(t₀)·D¹h(x₀) =

= u⁰(t₀)f⁰(t₀)²ⁿ⁻¹−(2n−1)u(t₀)f⁰(t₀)²ⁿ⁻²f⁰⁰(t₀)

f⁰(t₀)⁴ⁿ⁻² · 1

f⁰(t₀) =

= u⁰(t₀)·f⁰(t₀)−(2n−1)u(t₀)f⁰⁰(t₀)

f⁰(t0)²ⁿ⁺¹ = W(t₀) f⁰(t₀)^2(n+1)−1. Here

W(t0) =

= n

X

i=1

∂P_n

∂Xi

(A_n(t₀))Dⁱ⁺¹f(t₀) +

n

X

j=1

∂P_n

∂Yj

(A_n(t₀))D^j+1g(t₀)

D¹f(t₀)−

−(2n−1)Pn(An(t0))D²f(t0) =Pn+1(An+1(t0)).

We succeeded in proving the implication DⁿF(x₀) = P_n(A_n(t₀))

f⁰(t₀)²ⁿ⁻¹ ⇒Dⁿ⁺¹F(x₀) =P_n+1(A_n+1(t₀)) f⁰(t0)^2(n+1)−1 and the induction proof is complete.

Verification. Using the previous formulae forP₁,P₂,P₃ we get D¹F(x0) = g⁰(t0)

f⁰(t₀),

D²F(x₀) = f⁰(t₀)g⁰⁰(t₀)−f⁰⁰(t₀)g⁰(t₀) f⁰(t₀)³ , D³F(x₀) =

= f⁰(t0)²g⁰⁰⁰(t0) + 3f⁰⁰(t0)²g⁰(t0)−f⁰(t0)f⁰⁰⁰(t0)g⁰(t0)−3f⁰(t0)f⁰⁰(t0)g⁰⁰(t0)

f⁰(t₀)⁵ .

Remarks. 1. We have

∂

∂Xi(X₁^α1. . . X_n^αnY₁^β1. . . Yn^βn) = either 0 (in caseαi= 0) or αiX₁^α1. . . X_i^αi−1. . . X_n^αnY₁^β1. . . Y_n^βn (in caseαi6= 0)

∂

∂Yj(X₁^α1. . . X_n^αnY₁^β1. . . Y_n^βn) = either 0 (in caseβ_j = 0) or βjX₁^α1. . . X_n^αnY₁^β1. . . Y_j^βj−1. . . Yn^βn (in caseβj 6= 0).

We have P1(X1, Y1) =Y1, henceP1 is 1-degree homogeneous.

These facts, toghether with the recurrence formula (ii), imply that either P_n = 0 for some n (hence for all m ≥n) or P_n is n-degree homogeneous for all n. But Pn 6= 0 for any n, because one can take f :R → R,f(t) = t and g:R→R,g(t) =tⁿ,n∈N^∗henceF(x) =xⁿandDFⁿ(x)6= 0, which implies (see formula (∗ ∗ ∗))P_n is not null.

Conclusion: all Pn aren-degree homogeneous polynomials.

2. The third formula is easily used in order to compute succesively D¹F(x0),D²F(x0), . . . , DⁿF(x0), becauseP1, P2, . . . , Pnare easily computed.

Unfortunately, we have not a general formula forP_n.

3. In order to obtain a recurrence for computing Dⁿh(x₀), we have to work in the particular case g(t) ≡ t. Hence, one can take in the recurrence formula (ii): Y1= 1 andYk= 0, fork≥2 (becauseD¹g(t) = 1 andD^kg(t) = 0 for k≥2).

We get

Dⁿh(x0) = Pn(D¹f(t0), D²f(t0), . . . , Dⁿf(t0); 1,0,0, . . . ,0)

f⁰(t0)²ⁿ⁻¹ .

REFERENCES

[1] T.M. Apostol, Calculating higher derivatives of the inverses. Amer. Math. Month.107 (2000), 738–741.

[2] Cavaliere Francesco Fa`a di Bruno,Sullo svillupo delle funzioni. Annali di Scienze Mate- matiche e Fisiche6(1885), 479–480.

[3] Cavaliere Francesco Fa`a di Bruno,Note sur une nouvelle formule du calcul diff´erentiel.

Quarterly J. Pure Appl. Math.1(1857), 359–360.

[4] I. Chit¸escu, The Formula of Fa`a di Bruno and higher order derivatives of inverses.

Analele Univ. Buc. Mat.57(2008), 269–284.

[5] W.P. Johnson,The curious history of Fa`a di Bruno’s Formula. Amer. Math. Month.109 (2002), 217–234.

[6] W.P. Johnson,Combinatorics of higher derivatives of inverses. Amer. Math. Month.109 (2002), 273–277.

[7] S. Roman,The Formula of Fa`a di Bruno. Amer. Math. Month.87(1980), 805–809.

Received 26 April 2012 University of Bucharest

Faculty of Mathematics and Computer Science Academiei Str. 14

010014 Bucharest, Romania [email protected]