Definition of Tangent Space
Product rule is also used in definition of abstract tangent space of some abstract geometric figure (smooth manifold). This definition we can use if we cannot or wish to not use surrounding ambient space where our chosen geometric figure lives (since there might be no such surrounding space). It uses the fact that it is possible to define derivatives of real-valued functions on that geometric figure at a point p solely with the product rule and that the set of all such derivations in fact forms a vector space that is the desired tangent space.
Read more about this topic: Product Rule, Applications
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