Calcule
, sendo:
.
Passo 1
O enunciado pede para calcular as derivadas parciais de uma função composta usando a regra da cadeia. A função é
, com
e
. Vamos precisar da definição de derivada parcial, da regra da cadeia e das derivadas de
e de quocientes.
Passo 2
Primeiro calculamos as derivadas parciais de
e
em relação a
e
:
$$
\frac{\partial x}{\partial r}=2r,
\frac{\partial x}{\partial r}=2r,
\frac{\partial x}{\partial s}=-2s,
\frac{\partial y}{\partial r}=3r^2,
\frac{\partial y}{\partial s}=3s^2.
$$
Passo 3
Agora derivamos
parcialmente em relação a
e a
. Como
com
, temos:
$$
\frac{\partial w}{\partial x}=\frac{1}{y},e^{x/y},
\frac{\partial w}{\partial x}=\frac{1}{y},e^{x/y},
\frac{\partial w}{\partial y}=-\frac{x}{y2},e{x/y}.
$$
Passo 4
Aplicamos a regra da cadeia:
$$
\frac{\partial w}{\partial r}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial r},
\frac{\partial w}{\partial r}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial r},
\frac{\partial w}{\partial s}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial s}.
$$ Substituindo as derivadas encontradas:
$$
\frac{\partial w}{\partial r}=\frac{e{x/y}}{y}(2r)+\Bigl(-\frac{x}{y2}e{x/y}\Bigr)(3r2),
\frac{\partial w}{\partial r}=\frac{e{x/y}}{y}(2r)+\Bigl(-\frac{x}{y2}e{x/y}\Bigr)(3r2),
\frac{\partial w}{\partial s}=\frac{e{x/y}}{y}(-2s)+\Bigl(-\frac{x}{y2}e{x/y}\Bigr)(3s2).
$$
Passo 5
Por fim, escrevemos explicitando
e
e fatorando
:
$$
\frac{\partial w}{\partial r}=e{\frac{r2-s2}{r3+s3}}\Bigl(\frac{2r}{r3+s3}-\frac{3r2(r2-s2)}{(r3+s3)^2}\Bigr),
\frac{\partial w}{\partial r}=e{\frac{r2-s2}{r3+s3}}\Bigl(\frac{2r}{r3+s3}-\frac{3r2(r2-s2)}{(r3+s3)^2}\Bigr),
\frac{\partial w}{\partial s}=e{\frac{r2-s2}{r3+s3}}\Bigl(-\frac{2s}{r3+s3}-\frac{3s2(r2-s2)}{(r3+s3)^2}\Bigr).
$$