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\exametitle{Derivada de funcao composta de 2 variaveis 10 A}{Smirnov\mbox{}}{\today}

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\subsection*{Question}
\noindent Seja $f(x,y)=\arctan \left(y+\sqrt{x}\right)\,\log \left(y^4+x\right)$. Calcule as derivadas parciais
$\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ e $\frac{\partial^2 f}{\partial x\partial y}$.
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\subsection*{Sugestion}
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\subsection*{Resolution}
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Utilizando regra de cadeia obtemos
$$
\frac{\partial f}{\partial x} = {{\log \left(y^4+x\right)}\over{2\,\sqrt{x}\,\left(\left(y+\sqrt{x}
 \right)^2+1\right)}}+{{\arctan \left(y+\sqrt{x}\right)}\over{y^4+x}},
$$
$$
\frac{\partial f}{\partial y} = {{\log \left(y^4+x\right)}\over{\left(y+\sqrt{x}\right)^2+1}}+{{4\,
 y^3\,\arctan \left(y+\sqrt{x}\right)}\over{y^4+x}},
$$
$$
  \frac{\partial^2 f}{\partial x\partial y} = {{\left(-y-\sqrt{x}\right)\,\log \left(y^4+x\right)}\over{\sqrt{x}
 \,\left(\left(y+\sqrt{x}\right)^2+1\right)^2}}-{{4\,y^3\,\arctan 
 \left(y+\sqrt{x}\right)}\over{\left(y^4+x\right)^2}}+{{2\,y^3}\over{
 \sqrt{x}\,\left(y^4+x\right)\,\left(\left(y+\sqrt{x}\right)^2+1
 \right)}}+{{1}\over{\left(y^4+x\right)\,\left(\left(y+\sqrt{x}
 \right)^2+1\right)}} .
$$
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\subsection*{Result}
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$$ {{\log \left(y^4+x\right)}\over{2\,\sqrt{x}\,\left(\left(y+\sqrt{x}
 \right)^2+1\right)}}+{{\arctan \left(y+\sqrt{x}\right)}\over{y^4+x}}  $$
 $$ {{\log \left(y^4+x\right)}\over{\left(y+\sqrt{x}\right)^2+1}}+{{4\,
 y^3\,\arctan \left(y+\sqrt{x}\right)}\over{y^4+x}}  $$
 $$ {{\left(-y-\sqrt{x}\right)\,\log \left(y^4+x\right)}\over{\sqrt{x}
 \,\left(\left(y+\sqrt{x}\right)^2+1\right)^2}}-{{4\,y^3\,\arctan 
 \left(y+\sqrt{x}\right)}\over{\left(y^4+x\right)^2}}+{{2\,y^3}\over{
 \sqrt{x}\,\left(y^4+x\right)\,\left(\left(y+\sqrt{x}\right)^2+1
 \right)}}+{{1}\over{\left(y^4+x\right)\,\left(\left(y+\sqrt{x}
 \right)^2+1\right)}} $$
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\subsection*{Obs}




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