The derivative of a function Heydar Zahedani CSU San Marcos First the derivative was used, then discovered, and only then, defined. Fermat (1637-38) Given a line, divide it into two parts so that the product of the parts will be a maximum. ............ A....... | .......... B A................. B First solution: A( B A) AB A2 Second solution: (1)

( A h)( B ( A h)) AB A2 2 Ah Bh h 2 (2) Let (1) (2) AB A2 AB A2 2 Ah Bh h 2 2 Ah h 2 Bh (2 A h)h Bh h h 2 A h B Suppress h, A B 2 Note that it is the same as f ( A) A2 AB lim h 0

f ( A h) f ( A) f ( A) 0 h , where Fermat, Descartes, used the above method to solve the tangent problem Descartes 1637-38 Newton (1669) Newton and Leibniz independently invented calculus in the latter third of the seventeenth century Newton called our derivative a rate of flux or change and Leibniz called it the differential quotient So the derivative was discovered Leibniz (1684)

He claimed that any analytic function had a power series expansion LaGrange (1797) Cauchy (1823) In 1823 Cauchy defined the derivative as the limit of the quotient difference f ( x) lim h 0 f ( x h) f ( x ) h He proved the mean value theorem and gave the first proof of the existence of a solution to a differential equation There is everywhere continuous, nowhere differentiable function. Weierstrass (1861) How did the concept of derivative develop? Fermat implicitly used it, Newton and Leibniz discovered it, Taylor, Euler, Maclaurin developed it, LaGrange named and characterized it, only at the end of this long period (over 200 years) did Cauchy and

Weierstrass define it. Judith Grabinar Caratheodorys Theorem [ ] Let f be defined on an interval I containing the point c. Then f is differentiable at c if and only if there exists function on I that is continuous at c and sat isfies (1) f ( x) f (c ) ( x)( x c ) for xI In this case, we have (c) f (c). Proof. If f (c) exists, we can define by

Caratheodory (1953) f ( x ) f (c ) , x c ( x) x c f (c), x c. Then lim x c ( x) f (c) and so is continuous at c and for all x in I . Conversely if a function exists. Then (c) lim ( x) lim x c f (c) (c). x c f ( x ) f (c ) x c

f ( x ) f (c) ( x)( x c) that is continuous at c and satisfying (1) exists and so f is different iable at c and Caratheodorys definition of the derivative [ ] Let f be defined on an interval I containing the point c. Then f is differentiable at c if and only if there exists function on I that is continuous at c and satisfies (1) f ( x) f (c ) ( x)( x c) for xI In this case, we have (c) f (c). Note that if f (c) exists Example. Suppose c

then f is continuous at c. f ( x) x 3 for any real number x . Then for any number Caratheodory (1953) x3 c 3 ( x c)( x 2 cx c 2 ) Then ( x) x 2 cx c 2 is continuous at c and satisfies (1). So f is differentiable at c and Similarly, if f ( x) x n , then f (c) (c) 3c 2 . f ( x) nx n 1 Chain Rule Let I , J be intervals in R , let g : I R and f : J R be functions such that f ( J ) I . and let c J . If f is differentiable at c and g is differentiable at f (c) , then the composition function g f is differentiable at c and ( g f )(c ) g ( f (c)) f (c) Proof. f (c ) exists

f ( x) there is a function on J that is continuous at c and f (c) ( x)( x c) for x J where (c) f (c). g ( f (c)) exists there is a function defined on I such that is continuous at d f (c) and g ( y ) g (d ) ( y )( y d ) for y I , where (d ) g (d ). so g ( f ( x )) g ( f (c )) ( f ( x))( f ( x) f (c)) ( f ( x)[ ( x)( x c)] [( f )( x) ( x)]( x c) For all xJ such that f ( x) I . Note that ( f ) is continuous at c and [( f ) ](c) ( f (c )) (c) g ( f (c )) f (c)

Therefore g f is differentiable at c and ( g f )(c) g ( f (c)) f (c) . Caratheodory (1953) Frchet derivative Frchet derivative Let f : R n R m and a R n . f transformation Is differentiable at a if there exists a linear : R n R m such that lim x a f ( x ) f (a ) ( x a) x a 0

(2) Note that the definition can be extended for functions that are defined on Banach spaces. We can extend caratheodory definition to functions show that it is equivalent to Frchet derivative. f : R n R m and Definition Let f : R n R m and a R n . f : Rn Is differentiable at a if there exists a function M mn which is continuous at a and satisfies f ( x) f (a ) ( x )( x a) (1) Frchet 1878-1973

Theorem Every Frchet differentiable function is Caratheodory differentiable and vice versa. Proof Suppose (1) is given. Then f ( x) f (a) ( a)( x a) x a ( ( x) (a))( x a) x a ( x) ( a ) Since is continuous at a then lim f ( x) f ( a ) ( a)( x a) x a x a 0 .

So (1) implies (2) Suppose (2) is given and exists. Define by Frchet 1878-1973 1 {( f ( x) f ( a) ( x a)) ( x a) , x a 2 ( x) x a , x a. It is easily seen that ( x)( x a) f ( x) f (a) .(Note that product. (u v) w (u w)u, u R n , v, w R m ) Note also that at a. ( x) (a)

f ( x ) f (a) ( x a ) x a represents tensor and by (2) is continuous Basic Differen tiation theorems in Rn 1) If f , g : R n R m are differentiable at a R n , then f g , , R is also differentiable at a and D( f g )(a) Df (a) Dg (a) ( f g )( x) ( f ( a) g (a)) ( f ( x) f ( a)) ( g ( x ) g ( a) ( )( x)( x a) Since is continuous at a, it follows that D( f g )( a) Df (a) Dg ( a) 2) Chain rule If f is differentiable at and g is differentiable at differentiable at a and f (a) then

g f D( g f )(a) Dg ( f (a)) Df (a) Proof g ( f ( x ) g ( f (a ) ( f ( x))( f ( x) f (a )) ( f ( x )) ( x)( x a ) Note and f are continuous at a and Therefore D( g f )(a ) ( f (a )) (a) Dg ( f (a )) Df (a ) is continuous at f (a). is Frchet 1878-1973 Derivation on Algebras denote the algebra of all n by n complex matrices with respect to matrix multiplication. A derivation on M n (C ) is a linear mapping D : M n (C ) M n (C ) satisfying the product rule:

Let M n (C ) D( A B ) D ( A) D( B ) D(cA) cD ( A) D( AB) D( A) B AD( B ) For a fix matrix A in DA the mapping DA : M n (C ) M n (C ) that is defined by is a derivation and every derivation on for some A in M n (C ) (1942 Hochschid). DA ( X ) AX XA form M n (C )

M n (C ) is of the Barry Edward Johnson 1937-2002 If A B( H ) is an algebra of bounded operators on a Hibert Space then Every derivation on A is continuous and is of the form x ax xa for some a.(1966-Sakai, Kadison) Barry enjoyed lecturing at all levels from service teaching, even first year agriculture students, to postgraduate students and would initially describe himself to total outsiders as a 'teacher' References 1. Constantin Caratheodory, Theory of Functions of a Complex Variable, vol. 1, Chelsea, New York, 1954 2. Bartle, R. G. and D. R. Sherbet, Introduction to Real Analysis, 4 th ed , John Wiley& Sons, New York 2010 3. Stephen Kuhn, The derivative a la caratheodory, American Mathematical Monthly, Vol. 98, No.1, January 1991 4. Ernest Acosta and Cesar Delgado, Frchet vs. Caratheodory, American Mathematical Monthly, Vol. 101, No. 4, April 1994 5. Judith Grainer, The Changing Concept of Change, Mathematics Magazine, Vol.56, No.4 , 1983