"The mark of your ignorance is the depth of your belief in injustice and tragedy. What the caterpillar calls the end of the world, the Master calls the butterfly." ~ Richard Bach
I'm not sure what you're looking for here. There are Frechet derivatives and Fredholm operators, but these are concepts from functional analysis and are probably beyond the call of what you're looking for.
In my study of mathematics, I've never encountered "comparative differentiation," or differentiation with shapes. An Euclidean derivative (the ones encountered in calculus I-III) is a measure of the rate of change of a curve. Even at the level of vector calculus, you are still studying rates of change though now you are doing so in three dimensions.
As Snoink asked: Could you give us an example problem?
Best,
Brad
"If I have not seen as far as others, it is because giants were standing on my shoulders." -Hal Abelson
To have more, you have to become more. Don't wish it was easier - wish you were better. For things to change, you have to change, and for things to get better, you have to get better. — Jim Rohn
