Problema 1.34

kill(f)$
f(x,y):=x^3*y^3/(x^2+y^2);

diff(f(x,y),x);

ratsimp(%);

factor(%);

diff(f(x,y),y),factor;

diff(f(x,y),x,2),factor;

diff(f(x,y),y,2),factor;

diff(f(x,y),y,1,x,1),factor;

Problema 1.35  (a,-a) x+y distinto de 0

kill(f)$
f(x,y):=x*y*sin( (1/2)*(x-y)/(x+y) );

define(fx(x,y), (diff(f(x,y),x)));

x*y*(1/(2*(y+x))-(x-y)/(2*(y+x)^2));

ratsimp(%);

define(fy(x,y), factor(diff(f(x,y),y)));

ratsimp(%);

fy(x,0); fy(x,0)/x;

fx(0,y);  fx(0,y)/y;  

diff(f(x,y),x,1,y,1), factor;

ev(%,y=m*x);

ratsimp(%);

diff(f(x,y),x,2), factor;

diff(f(x,y),y,2), factor;

ev(%,y=m*x);

ratsimp(%);

Problema 1.36

kill(f)$
f(x,y):=x*y*(x^2-y^2)/(x^2+y^2);

define(fx(x,y), factor(diff(f(x,y),x)));

define(fy(x,y), factor(diff(f(x,y),y)));

fy(x,0); fy(x,0)/x;

fx(0,y);  fx(0,y)/y;  

diff(f(x,y),x,1,y,1), factor;

ev(%,y=m*x);

ratsimp(%);

diff(f(x,y),x,2), factor;

diff(f(x,y),y,2), factor;

ev(%,y=m*x);

ratsimp(%);

Problema 1.37

kill(f)$
f(x,y):=y^2*(sin(x/y));

define(fx(x,y), factor(diff(f(x,y),x)));

define(fy(x,y), factor(diff(f(x,y),y)));

diff(f(x,y),x,1,y,1), factor;

ev(%,y=m*x);

ratsimp(%);

diff(f(x,y),x,2), factor;

ev(%,y=m*x);

diff(f(x,y),y,2), factor;

ev(%,y=m*x);

ratsimp(%);

problema 1.39

f(x,y):= (sin(x*y)-x*y)/(x^2*y);

fx:diff(f(x,y),x);

fx1:first(fx),factor;

fx2:second(fx),factor;

fx1+fx2-fx,ratsimp;

fx1+fx2;

fy:diff(f(x,y),y);

fy1: first(fy),factor;

fy2:second(fy),factor;

fy1+fy2-fy,ratsimp;

fx1+fx2;

diff(f(x,y),x);

ratsimp(%);

fxy:diff(f(x,y),x,1,y,1),ratsimp;

fxy:diff(f(x,y),y,1,x,1),ratsimp;

diff(f(x,y),x,1,y,1);

-(y*cos(x*y)-y)/(x^2*y^2)-(2*(x*cos(x*y)-x))/(x^3*y)

ratsimp(-(y*cos(x*y)-y)/(x^2*y^2)-(2*(x*cos(x*y)-x))/(x^3*y));