The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes' theorem.

Let ${\displaystyle G}$ be a Lie group, and let ${\displaystyle {\mathfrak {g}}}$ be its Lie algebra. The Maurer-Cartan form, ${\displaystyle \omega _{G}}$, is the smooth ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle 1}$-form on ${\displaystyle G}$ (cf. Lie algebra valued form) defined by

for all ${\displaystyle g\in G}$ and ${\displaystyle X_{g}\in T_{g}G}$. Here ${\displaystyle L_{g}}$ denotes left multiplication by the element ${\displaystyle g\in G}$ and ${\displaystyle T_{g}L_{g}}$ is its derivative at ${\displaystyle g}$.

Let ${\displaystyle f:M\to G}$ be a smooth function between a smooth manifold ${\displaystyle M}$ and ${\displaystyle G}$. Then the Darboux derivative of ${\displaystyle f}$ is the smooth ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle 1}$-form

the pullback of ${\displaystyle \omega _{G}}$ by ${\displaystyle f}$. The map ${\displaystyle f}$ is called an integral or primitive of ${\displaystyle \omega _{f}}$.

The reason that one might call the Darboux derivative a more natural generalization of the derivative of single-variable calculus is this. In single-variable calculus, the derivative ${\displaystyle f'}$ of a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ assigns to each point in the domain a single number. According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a linear map from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates two pieces of data: the image of the domain point and the linear map. In single-variable calculus, we drop some information. We retain only the linear map, in the form of a scalar multiplying agent (i.e. a number).

One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of ${\displaystyle \mathbb {R} }$ under addition. The tangent bundle of any Lie group can be trivialized via left (or right) multiplication. This means that every tangent space in ${\displaystyle \mathbb {R} }$ may be identified with the tangent space at the identity, ${\displaystyle 0}$, which is the Lie algebra of ${\displaystyle \mathbb {R} }$. In this case, left and right multiplication are simply translation. By post-composing the manifold-type derivative with the tangent space trivialization, for each point in the domain we obtain a linear map from the tangent space at the domain point to the Lie algebra of ${\displaystyle \mathbb {R} }$. In symbols, for each ${\displaystyle x\in \mathbb {R} }$ we look at the map

Since the tangent spaces involved are one-dimensional, this linear map is just multiplication by some scalar. (This scalar can change depending on what basis we use for the vector spaces, but the canonical unit vector field ${\displaystyle {\frac {\partial }{\partial t}}}$ on ${\displaystyle \mathbb {R} }$ gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by ${\displaystyle f'(x)}$.