# Exercici 11.9

kill(all)$

f(x,y):= exp(x) + exp(y) ;
g1(t):=t^2$  g2(t):=log(t)$   g(t):=[g1(t) , g2(t)];

## Apartat a)

Funcio f: diferenciable en R2.
Funcio g: diferenciable en {x>0}

Jf(x,y) = Jf: jacobian([f(x,y)] , [x,y]) ;

Jg(t) = Jg: jacobian([g1(t) , g2(t)] , [t]) ;

## Apartat b)

define (F1(x,y) , g1(f(x,y)));
define (F2(x,y) , g2(f(x,y)));
F(x,y):= [F1(x,y) , F2(x,y)];

JF(x,y) = JF: jacobian([F1(x,y),F2(x,y)] , [x,y]) ;

define (G(t) , f(g1(t), g2(t)));

JG(t) = JG: jacobian([G(t)] , [t]) ;

'JF(1,0) = JF10 : subst([x=1, y=0] , JF);

'JG(1) = JG1 : subst([t=1] , JG);

## Apartat c)

G(1);

define (DG(t) , diff(G(t) , t));

Eq1: t = 1  + (1/DG(1))*(u - G(1));

ratsimp(%);