2.5.1 Regras da Cadeia

Teorema 13 (Regras da Cadeia). Se $w=f(u,v)$ , $u=g(x,y)$ e $v=h(x,y)$ e se $f$ , $g$ e $h$ são diferenciáveis então

$\dfrac {\partial w}{\partial x}=\dfrac {\partial w}{\partial u}\dfrac {\partial u}{\partial x}+\dfrac {\partial w}{\partial v}\dfrac {\partial v}{\partial x};$

$\dfrac {\partial w}{\partial y}=\dfrac {\partial w}{\partial u}\dfrac {\partial u}{\partial y}+\dfrac {\partial w}{\partial v}\dfrac {\partial v}{\partial y}.$

Prova: Incrementando $x$ e mantendo $y$ constante temos

$\Delta u=g(x+\Delta x, y)-g(x,y)$

$\Delta v=h(x+\Delta x, y)-h(x,y)$

$\Delta w=\dfrac {\partial w}{\partial u}\Delta u+\dfrac {\partial w}{\partial v}\Delta v+\varepsilon _1 \Delta u+\varepsilon _2 \Delta v$

onde $\lim _{(\Delta u, \Delta v)\to (0,0)} \varepsilon _ i=0.$

$\dfrac {\Delta w}{\Delta x}=\dfrac {\partial w}{\partial u}\dfrac {\Delta u}{\Delta x}+\dfrac {\partial w}{\partial v}\dfrac {\Delta v}{\Delta x}+\varepsilon _1 \dfrac {\Delta u}{\Delta x}+\varepsilon _2 \dfrac {\Delta v}{\Delta x}$

mas

$\dfrac {\partial w}{\partial x}=lim_{\Delta x \to 0}\dfrac {\Delta w}{\Delta x}, \, \dfrac {\partial u}{\partial x}=lim_{\Delta x \to 0}\dfrac {\Delta u}{\Delta x},\, \dfrac {\partial v}{\partial x}=lim_{\Delta x \to 0}\dfrac {\Delta v}{\Delta x}$

temos que

$\Delta u \to 0 \quad \text {quando} \quad \Delta x \to 0$

$\Delta v \to 0 \quad \text {quando} \quad \Delta x \to 0$

com isso, concluimos que

$\dfrac {\partial w}{\partial x}=\dfrac {\partial w}{\partial u}\dfrac {\partial u}{\partial x}+\dfrac {\partial w}{\partial v}\dfrac {\partial v}{\partial x}.$

de modo análogo se prova a outra parte do teorema.

Exemplo 34. Ache $\dfrac {\partial w}{\partial p}$ e $\dfrac {\partial w}{\partial q}$ se

$w=r^3+s^2, \quad r=pq^2, \quad s=p^2 \operatorname{sen}q.$

Solução: Temos que

$\dfrac {\partial w}{\partial p}=\dfrac {\partial w}{\partial r}\dfrac {\partial r}{\partial p}+\dfrac {\partial w}{\partial s}\dfrac {\partial s}{\partial p}$

$=3r^2q^2+2s2p\operatorname{sen}q=3p^2q^6+4p^3\operatorname{sen}^2 q$

$\dfrac {\partial w}{\partial q}=\dfrac {\partial w}{\partial r}\dfrac {\partial r}{\partial q}+\dfrac {\partial w}{\partial s}\dfrac {\partial s}{\partial q}$

$=3r^2 2pq+2sp^2 \cos q=3(pq^2)2pq+2(p^2 \operatorname{sen}q) p^2 \cos q$

$=6p^2q^3+2p^4\operatorname{sen}q \cos q.$

Exemplo 35. Use a regra da cadeia para achar $\dfrac {\partial w}{\partial z}$ se

$w=r^2+sv+t^3,$

com $r=x^2+y^2+z^2$ , $s=xyz$ , $v=x e^ y$ e $t=y z^2.$

Solução:

$\dfrac {\partial w}{\partial z}=\dfrac {\partial w}{\partial r}\dfrac {\partial r}{\partial z}+\dfrac {\partial w}{\partial s}\dfrac {\partial s}{\partial z}+\dfrac {\partial w}{\partial v}\dfrac {\partial v}{\partial z}+\dfrac {\partial w}{\partial t}\dfrac {\partial t}{\partial z}$

$=2r(2z)+vxy+s0+3t^22yz$

$4z(x^2+y^2+z^2)+x^2y e^ y+ 6y^3z^5.$

Se $w$ é função de duas variáveis e cada uma delas é função de uma variável $t$ , então $w$ é função de $t$ .

Exemplo 36. Seja $w=x^2+yz$ , onde $x=3t^2+1$ , $y=2t-4$ , $z=t^3$ , ache $dfrac{dw}{dt}.$

Temos pela regra da cadeia que

$\dfrac {dw}{dt}=\dfrac {\partial w}{\partial x}\dfrac {\partial x}{\partial t}+\dfrac {\partial w}{\partial y}\dfrac {\partial y}{\partial t}+\dfrac {\partial w}{\partial z}\dfrac {\partial z}{\partial t}$

$\dfrac {dw}{dt}=2x(6t)+z(2)+y(3t^2)=2(3t^2+1)+2t^3+3(2t-4)t^2.$

Exemplo 37.

Um circuito elétrico simples consiste em um resistor $R$ e uma força eletromotriz $V.$ Em certo instante, $V$ é de $80$ volts e aumenta à taxa de $5$ volts/min, enquanto $R$ é de $40$ ohms e decresce à razão de $2$ ohms/min. Use a lei de Ohm, $I=V/R$ , e uma regra da cadeia, para achar a taxa à qual a corrente $I$ (em ampéres) varia.