Theorem on the differentiation of a composite function with a vector argument

The paper provides a theorem on the differentiation of a composite function with a vector argument. The theorem shows how the partial derivative of the total derivative of the composite function can be expressed through the total derivative of the partial derivative of the composite function. The proof of the theorem is based on Mishkov’s formula, which is the generalization of the well-known Faà di Bruno’s formula for a composite function with a vector argument.


INTRODUCTION
The theorem proved in this paper was required as an intermediate result in solving the problem of the transformation of the nonlinear control system, described by state equations, into the observer form and finding the necessary conditions for the possibility of such transformation [2].The deduction of the necessary conditions involves frequent application of the differentiation of the composite functions with respect to time argument and taking the partial derivatives of the differentiated composite function with respect to one of the variables or its derivatives.The goal of this paper is to present and prove a formula (commutation rule) which allows changing the order of taking the total higher-order derivatives of the composite function and their partial derivatives with respect to one of the variables or its derivative.Since this result may be useful in the solution of other nonlinear control problems, we propose it as a separate contribution.For example, probably the main result, provided in the paper, can be applied for observer design in [4].
The main tool for proving the theorem (commutation rule) is Mishkov's theorem [3] which provides the explicit formula for the nth derivative of a composite function with a vector argument.Mishkov's formula is a straightforward generalization of the well-known Faà di Bruno's formula [1] which gives an explicit equation for the nth-order derivative of the composite function with a scalar argument.

MAIN RESULT
The following theorem shows how the partial derivative of the total derivative of the composite function can be expressed through the total derivative of the partial derivative of this function.The composite function with the vector argument with an arbitrary number of components is considered.
Theorem 1. Assume that f (ξ 1 (t), ξ 2 (t), . . ., ξ r (t)) is a composite function for which derivatives up to order a + b are defined; then , where l = 1, 2, . . ., r, C b a+b is the binomial coefficient and a, b are nonnegative integers.Proof.In the proof we omit the variable t of ξ i (t), i.e. use instead of ξ i (t) a shorter notation ξ i , which allows the bulky formulas to be written in a more compact form.According to Mishkov's formula [3], the (a + b)th derivative of the composite function with a vector argument can be computed by the formula where the respective sums are taken over all nonnegative integer solutions of the Diophantine equations as follows: for i = 1, . . ., a + b, and p j and k on the right-hand side of (1) satisfy the relations In taking the partial derivative of sum (1) with respect to ξ (a) l , only addends of sum (1) with q a,l = 0 will matter.Denote by h(•) and g(•) the parts of sum (1) corresponding to q a,l = 0 and q a,l = 0, respectively; then Note that it is possible to state that h(•) equals the expression in the right-hand side of (1) where, in addition to the restrictions expressed by ( 2), (3), and (4), the condition q a,l = 0 has to be satisfied.Note also that if q a,l = 0, then k a = 0. We prove the formula separately for the cases a > b and a ≤ b.First, consider the case when a > b.Since k a = 0 and q a,l = 0, in order to satisfy (2) and (3), the following must hold As a result, under the condition q a,l = 0, one can rewrite (2) as follows: and in (3), now i = 1, . . ., b.
Using (6) and changing the notations, taking p j = p j for j = 1, 2, . . ., r, j = l, pl = p l − 1 and k = k − 1, equations (4) may be rewritten as Note also that under conditions (6) and Taking into account the above equations and the fact that the partial derivative of g(•) in (5) with respect to ξ (a) l equals 0, we obtain, in new variables p j and k: Note that in (9) all the partial derivatives with respect to ξ j are of order p j except with respect to ξ l when the order of the partial derivative is pl + 1.In order to unify the orders, denote f := ∂ f ∂ ξ l .We also multiply the right-hand side of equation ( 9) by b! b! to obtain It is easy to observe now that, according to Mishkov's formula, the sum on the right-hand side of (10) together with the conditions (3) for i = 1, . . ., b, (7) and (8), is the bth-order total derivative of the function f .Consequently, Second, consider the case a ≤ b.Since k a = 0, in order to satisfy (2) and (3), the following must hold: Therefore, it is possible to rewrite condition (2) as and in ( 3), now i = 1, . . ., b. Again, in order to unify the notation in (13), one can take ki = k i for i = 1, 2, . . ., b, i = a and ka = k a − 1.This allows (13) to be rewritten as follows: and (3) as Since q a,l ≥ 1, we can denote qa,l := q a,l − 1 and the remaining q's as qi, j := q i, j .Thereby (15) can be rewritten in unified notation as for i = 1, . . ., b. Changing notations, taking p j = p j for j = 1, 2, . . ., r, j = l, pl = p l − 1 and k = k − 1, equations ( 4) may be rewritten as Taking (12) into account and using variables ki and qi, j , we have Furthermore, on the basis of (18) and the fact that the partial derivative of g(•) in (5) with respect to ξ (a) l equals 0, we obtain, in new variables p j and k Like in case a > b we denote f = ∂ f ∂ ξ l and multiply the right-hand side of the equality given above by b! b! to obtain Again it is not difficult to observe that according to Mishkov's formula, the sum on the right-hand side of equation ( 19), together with the conditions ( 14), ( 16), and (17), is the bth-order total derivative of the function f .Consequently, (11) holds again, and this completes the proof.
Some useful corollaries of the theorem are given below.
Corollary 1.Under the assumptions of Theorem 1 , where m and n are nonnegative integers.
Corollary 2. Under the assumptions of Theorem 1 where n is a nonnegative integer.

EXAMPLE
The example in this section illustrates the statement of Theorem 1.Consider the composite function f (x(t), y(t)) and assume that we need to take the partial derivative with respect to ÿ(t) of the 3rd-order total derivative of the function.Direct computations yield On the other hand, taking the partial derivative of f (x(t), y(t)) with respect to y(t) and the total derivative of the obtained result, one gets Multiplying both sides of the above equality by C .

CONCLUSIONS
The paper shows how to commute the operations of taking higher-order total and partial derivatives of composite functions with vector arguments.The formula, provided in the paper, may be applicable not only in differential calculus.As already mentioned in the introduction, the theorem was a useful tool in deriving solvability conditions of a certain problem in nonlinear control theory.With high probability it may be useful in dealing with other nonlinear control problems where the derivatives of the composite functions with a vector argument often show up.

ACKNOWLEDGEMENT
This work was partially supported by the Estonian Science Foundation through grant No. 6922.