Regra da Cadeia — Formas Clássicas
Corolário. Se $f: \mathbb{R}^n \to \mathbb{R}^m$ e $g: \mathbb{R}^m \to \mathbb{R}^p$, com componentes,
$$\frac{\partial (g \circ f)i}{\partial x_j}(a) = \sum{k=1}^{m} \frac{\partial g_i}{\partial y_k}(f(a)) \cdot \frac{\partial f_k}{\partial x_j}(a).$$
Exemplo 1. Sejam $u = x^2 - y^2$, $v = 2xy$, e $g(u,v) = u^2 + v^2$. Então $g(f(x,y)) = (x^2-y^2)^2 + 4x^2y^2 = (x^2+y^2)^2$. Pela cadeia:
$$\frac{\partial g}{\partial x} = 2u \cdot 2x + 2v \cdot 2y = 4x(x^2-y^2) + 4y(2xy) = 4x(x^2+y^2).$$
De fato, $\frac{\partial}{\partial x}(x^2+y^2)^2 = 4x(x^2+y^2)$. $\checkmark$
Exemplo 2. Se $f(t) = (t^2, e^t)$ e $g(x,y) = x\sin(y)$, então
$$\frac{d}{dt}g(f(t)) = \frac{\partial g}{\partial x}\cdot 2t + \frac{\partial g}{\partial y}\cdot e^t = \sin(e^t)\cdot 2t + t^2\cos(e^t)\cdot e^t.$$
Exemplo 3. Para $z = f(x,y)$ com $x = r\cos\theta$, $y = r\sin\theta$:
$$\frac{\partial z}{\partial r} = \frac{\partial f}{\partial x}\cos\theta + \frac{\partial f}{\partial y}\sin\theta, \qquad \frac{\partial z}{\partial \theta} = -\frac{\partial f}{\partial x}\, r\sin\theta + \frac{\partial f}{\partial y}\, r\cos\theta.$$