Say I have a matrix
$$ \mathbf{M}(a,b,c,d)=\pmatrix{a&b\\c&d} $$
Its exponential is
$$ f(a,b,c,d)=\exp \mathbf{M}(a,b,c,d) $$
The partial derivative, obtained by the first order Taylor expansion in a,b,c,d is
$$ f(a+\Delta a,b,c,d)\approx f(a,b,c,d)+ \Delta a \pmatrix{1 &0\\0&0} \exp \mathbf{M}(a,b,c,d)\\ \implies \frac{\partial f(a,b,c,d)}{\partial a}=\pmatrix{1 &0\\0&0} \exp \mathbf{M}(a,b,c,d) $$
Finally, the total derivative is:
$$ df(a,b,c,d)=da\pmatrix{1 &0\\0&0} \exp \mathbf{M}(a,b,c,d)+db\pmatrix{0 &1\\0&0} \exp \mathbf{M}(a,b,c,d)+\dots\\ =\pmatrix{da\exp \mathbf{M}(a,b,c,d) & db \exp \mathbf{M}(a,b,c,d) \\ dc \exp \mathbf{M}(a,b,c,d) & dd \exp \mathbf{M}(a,b,c,d)} $$
What is the geometric or mathematical meaning of this result. What is the geometric meaning of the partial derivative or a total derivative of a matrix exponential?
For example of what I am looking for; The total derivative of a function can be interpreted as an infinitesimal displacement of a function (like a line for instance). can the total derivative of the exponential of a matrix be interpreted as a line in "general linear space" - does it relate to an evolution of the general linear group?