#title: Derivada de funcao composta de 2 variaveis 10 A

#author: Smirnov

#let:

## f def_e = e :: %e;
f F  = [%e^u, sin(u), cos(u), tan(u), atan(u)];
f G  = [v^2, v^3, sqrt(v), log(v)];
f u1 = [x^2+y, sqrt(x)+y, x+sqrt(y)];
f v1 = [x+y^4, sqrt(x)+y, sqrt(x+y)];

maxima{ FF(u) :=#F ;
       GG(v)  :=#G ;
       H(u,v) :=FF(u)*GG(v);
       u2(x,y):=#u1; 
       v2(x,y):=#v1;   };

f HH = H(u2(x,y),v2(x,y));

f Dx = diff(#HH,x,1);
f Dy = diff(#HH,y,1);
f Dxy = diff(#HH,x,1,y,1);

#question:

\noindent Seja $f(x,y)=#HH$. Calcule as derivadas parciais
$\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ e $\frac{\partial^2 f}{\partial x\partial y}$.

#sugestion:



#resolution:  Utilizando regra de cadeia obtemos
$$
\frac{\partial f}{\partial x} = #Dx,
$$
$$
\frac{\partial f}{\partial y} = #Dy,
$$
$$
  \frac{\partial^2 f}{\partial x\partial y} = #Dxy .
$$


#result:
 $$ #Dx  $$
 $$ #Dy  $$
 $$ #Dxy $$


#verification

 #Dx  type=x_xmaxfun ;
 #Dy  type=x_xmaxfun ; 
 #Dxy type=x_xmaxfun ;
 
#fim