"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.
The author is a prof at the Indian Statistical Institute (Kolkata)
This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms.
I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.
"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it.