This contribution outlines the physics potential of a future muon collider operating at a center‑of‑mass energy of 10 TeV for precision studies of the Higgs sector. A detailed detector simulation, incorporating the dominant sources of machine-induced background, is employed to assess the expected sensitivity to key Higgs processes. These include measurements of production cross sections for $H \to b\bar{b}$, $H \to WW^\ast$, and double-Higgs production $H\!H \to b\bar{b}b\bar{b}$. A central focus of the study is the determination of the Higgs boson trilinear self-coupling, a critical parameter for understanding the structure of the Higgs potential and the mechanism of electroweak symmetry breaking. The studies are based on the MUSIC (MUon System for Interesting Collisions) detector concept, specifically designed for the muon collider environment, and assumes two interaction points, each delivering an integrated luminosity of 10 ab$^{-1}$ over five years. The results demonstrate the exceptional prospects of a multi-TeV muon collider in probing the Higgs potential with a precision unattainable by any other proposed future collider within a comparable timeframe.