Notation and definitions
Calculus in higher dimensions
The derivative of a function \(f\) is the function \(Df\) whose value for a particular argument is something that can be multiplied by an increment \(Δx\) in the argument to get a linear approximation to the increment in the value of \(f\) (Sussman, Wisdom, and Farr 2013, 202): \[ f(x) ≈ f(x_0)+ Df(x_0)(x-x_0). \] The derivative at a point \(x\) must be a covector, in comparison to the gradient, which is defined as a vector.
Derivatives of compositions obey the chain rule: \[ D(f ∘ g) = ((Df) ∘ g) Dg. \] So at \(x\), \[ (D(f ∘ g))(x) = Df(g(x)) Dg(x). \] For example, \(D(f(x(t))) = Df(x(t)) Dx(t) = Df(x(t)) ẋ(t)\). One could also write it with subscripts denoting the point of evaluation: \[ D(f ∘ g)(a) = Df(g(a)) ∘ Dg(a). \]
Partial derivatives are just the components of the derivative of a function that takes multiple arguments. So a partial derivative of a function is a composition of a component selector and the derivative of that function (Sussman, Wisdom, and Farr 2013, 204). Indeed: \[ ∂_0 g = I_0 ◦ Dg, \\ ∂_1 g = I_1 ◦ Dg. \] We can thus write \[ Dg = [\partial_0 g, \partial_1 g, \dots]^T. \] Partial derivatives of compositions also obey a chain rule: \[ ∂_i(f ◦ g)=((Df) ◦ g) · ∂_i g. \] So if \(x\) is a tuple of arguments, then \[ (∂_i(f ◦ g))(x)=Df(g(x)) · ∂_ig(x). \]
Comparison to “traditional” notation from Wikipedia: Given functions \(f:\R^n → \R\), \(x:\R→ \R^n\) the chain rule is \[ \begin{aligned} \frac{\d}{\d t} f(x(t)) = \frac{\d}{\d t}f(x_1(t), \dots, x_n (t)) &= \sum_{i=1}^n \left(\frac{\d}{\d t}{x_i}(t)\right) \partial_i f(x_1(t), \dots, x_n (t)) \\ &= \sum_{i=1}^n \dot x_i(t) \partial_i f(x(t)) \\ &= \sum_{i=1}^n \dot x_i(t) \partial_i f(x(t)) \left[ = Df(x(t)) Dx(t)\right] \end{aligned} \]