When we learn derivative in calculus, we know that derivative at point is number. When we move to vector function we understand that derivative is linear operator. This allows use Jacobian as expression for derivative. However in this case we again have option to separate derivative and increment of argument.

When we start to work with function of division ring (quaternion valued function for instance) we expect something similar. However it does not work or makes set of differentiable functions very limitted. Derivative of function of division ring is not number, but additive map and it is easy to see that this is the Gateaux derivative. When we learn derivative of function of division ring we cannot separate derivative itself and increment of argument.

For instance derivative of function

*f(x)=axb*has form*∂f(x)(h)=ahb*

Derivative of function

*f(x)=x*^{2 }has form*∂f(x)(h)=xh+hx*

Derivative of function

*f(x)=x*has form^{-1}*∂f(x)(h)=-x*

^{-1}hx^{-1}