Computing the linear response, or the derivative of long-time-averaged observables with respect to system parameters, is a central problem for many applications. Conventionally, there are three linear response formulas: the path-perturbation formula (including the backpropagation method in machine learning), the divergence formula, and the probability-kernel-differentiation formula. But none works for the general case, which is chaotic, high-dimensional, and small-noise. Then we present our fast response formula for hyperbolic systems, expressed by a pointwisely defined function; some of our ideas are from the classic proof of the hyperbolic linear responses. Hence, people can compute the linear response by sampling, that is, compute the average of some function over an orbit. The fast response formula overcomes all three difficulties under hyperbolicity assumptions. Then we discuss how to further incorporate kernel-differentiation to overcome non-hyperbolicity.