21-26 Utilize a Regra da Cadeia para determinar as derivadas parciais indicadas.
N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v ; \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \quad quando u=2 \quad v=3, \quad w=4

Passo 1

Resumindo o enunciado: temos
N = \frac{p + q}{p + r},
onde
p = u + v w,\quad q = v + u w,\quad r = w + u v.
O conteúdo é aplicação da Regra da Cadeia, pois N depende de p,q,r e cada um destes depende de u,v,w. Vamos calcular as derivadas parciais \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v} e \frac{\partial N}{\partial w} e avaliar em (u,v,w)=(2,3,4).

Passo 2

Primeiro, derivamos N em relação a p,q,r, tratando q e r como constantes quando derivamos em relação a p, etc.
N = \frac{p+q}{p+r}\quad\Longrightarrow
\frac{\partial N}{\partial p} = \frac{(1)(p+r)-(p+q)(1)}{(p+r)^2} = \frac{r-q}{(p+r)^2},
\frac{\partial N}{\partial q} = \frac{1}{p+r},
\frac{\partial N}{\partial r} = -\,\frac{p+q}{(p+r)^2}.

Passo 3

Agora obtemos as derivadas de p,q,r em relação a u,v,w:
p = u + v w\;\Longrightarrow\; \frac{\partial p}{\partial u} = 1,\quad \frac{\partial p}{\partial v} = w,\quad \frac{\partial p}{\partial w} = v,
q = v + u w\;\Longrightarrow\; \frac{\partial q}{\partial u} = w,\quad \frac{\partial q}{\partial v} = 1,\quad \frac{\partial q}{\partial w} = u,
r = w + u v\;\Longrightarrow\; \frac{\partial r}{\partial u} = v,\quad \frac{\partial r}{\partial v} = u,\quad \frac{\partial r}{\partial w} = 1.

Passo 4

Aplicamos a Regra da Cadeia em cada variável. Por exemplo,
\frac{\partial N}{\partial u} = \frac{\partial N}{\partial p}\,\frac{\partial p}{\partial u} + \frac{\partial N}{\partial q}\,\frac{\partial q}{\partial u} + \frac{\partial N}{\partial r}\,\frac{\partial r}{\partial u}.
Analogamente,
\frac{\partial N}{\partial v} = \frac{\partial N}{\partial p}\,\frac{\partial p}{\partial v} + \frac{\partial N}{\partial q}\,\frac{\partial q}{\partial v} + \frac{\partial N}{\partial r}\,\frac{\partial r}{\partial v},
\frac{\partial N}{\partial w} = \frac{\partial N}{\partial p}\,\frac{\partial p}{\partial w} + \frac{\partial N}{\partial q}\,\frac{\partial q}{\partial w} + \frac{\partial N}{\partial r}\,\frac{\partial r}{\partial w}.

Passo 5

Agora avaliamos em u=2, v=3, w=4. Primeiro calculamos
p = 2 + 3\cdot4 = 14,\quad q = 3 + 2\cdot4 = 11,\quad r = 4 + 2\cdot3 = 10,\quad p + r = 24.
Logo
\frac{\partial N}{\partial p} = \frac{r-q}{(p+r)^2} = \frac{10 - 11}{24^2} = -\frac{1}{576},
\frac{\partial N}{\partial q} = \frac{1}{24}, \quad \frac{\partial N}{\partial r} = -\,\frac{p+q}{24^2} = -\,\frac{25}{576}.
Agora:
\frac{\partial N}{\partial u} = \Bigl(-\tfrac{1}{576}\Bigr)\cdot1 + \tfrac{1}{24}\cdot4 + \Bigl(-\tfrac{25}{576}\Bigr)\cdot3 = \frac{5}{144},
\frac{\partial N}{\partial v} = \Bigl(-\tfrac{1}{576}\Bigr)\cdot4 + \tfrac{1}{24}\cdot1 + \Bigl(-\tfrac{25}{576}\Bigr)\cdot2 = -\,\frac{5}{96},
\frac{\partial N}{\partial w} = \Bigl(-\tfrac{1}{576}\Bigr)\cdot3 + \tfrac{1}{24}\cdot2 + \Bigl(-\tfrac{25}{576}\Bigr)\cdot1 = \frac{5}{144}.

Resposta

\frac{\partial N}{\partial u}\Big|_{(2,3,4)} = \frac{5}{144},\quad \frac{\partial N}{\partial v}\Big|_{(2,3,4)} = -\frac{5}{96},\quad \frac{\partial N}{\partial w}\Big|_{(2,3,4)} = \frac{5}{144}.

Exercícios relacionados