I know that the derivative of a function $f: \mathbf{R}^n \to \mathbf{R}^m$ at $x \in \mathbf{R}^n$ can be defined to be the linear function $T: \mathbf{R}^n \to \mathbf{R}^m$ satisfying $f(x + h) = f(x) + T(h) + E(h)$ where $\lim_{h \to 0} \frac{\| E(h) \|}{\| h \|} = 0$.
Obviously, we can modify this definition and require $T$ to possess any property we like. For example, we can require $T$ to be just additive instead of linear or we can require it to be orthogonal in addition to being linear.
Is there a name for this kind of generalization of derivatives and how can I find more information about them?