We study the scaling behavior of fidelity susceptibility density (χ_ f) at or close to an anisotropic quantum critical point characterized by two different correlation length exponents ν_|| and ν_ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an anisotropic quantum critical point is different from that seen near an isotropic quantum critical point. In particular, for a finite

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... m with linear dimension L_|| (L_) in the parallel (perpendicular) directions, the maximum value of χ_ f is found to increases in a power-law fashion with L_|| for small L_||, with an exponent depending on both ν_|| and ν_ and eventually crosses over to a scaling with L_ for L_||^1/ν_||≳ L_^1/ν_. We also propose scaling relations of heat density and defect density generated following a quench starting from an anisotropic quantum critical point and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.