# Rotation derivative in 3D

I have this seemingly easy question that I have been unable to answer myself for a lack of some basics or maybe just terminology.

Consider the function $f : \mathbb{R}^6 \rightarrow \mathbb{R}^3$ that takes for parameters a 3D point and the three Euler Angles $\alpha, \beta, \gamma$ and outputs a rotated 3D point. More specifically

$[\hat{x}, \hat{y}, \hat{z}]^T = f(x, y, z, \alpha, \beta, \gamma) = R(\alpha, \beta, \gamma) * [x, y, z]^T$

where $R(\alpha, \beta, \gamma)$ is the rotation matrix constructed from the Euler Angles.

I am interested to learn

• what the partial derivatives $\frac{\partial f}{\partial \alpha}, \frac{\partial f}{\partial \beta}, and \frac{\partial f}{\partial \gamma}$ are referred to as (do they have a name?)
• and if there is a smart way to evaluate them at a given point that doesn't involve differentiating the rotation matrix (in a manner of speaking) or other rotation formalisms such as rotation axes or quaternions.

• it should be $f' = (\frac{\partial f}{\partial x},\ldots)^T$ – user190080 Aug 15 '16 at 16:32