We derive the path-kernel formula for the linear response, the parameter derivative of averaged observables, of SDEs. Here the parameter controls initial conditions, drift coefficients, and diffusion coefficients. Our formula tempers the unstableness by gradually moving the path-perturbation to hit the probability kernel. It does not assume hyperbolicity but requires (either multiplicative or additive) noise. It extends the path-perturbation formula (or stochastic gradient method), the Bismut-Elworthy-Li formula, and a formula in Malliavin calculus (or likelihood ratio method). Then we derive a pathwise sampling algorithm and demonstrate it on the 40-dimensional Lorenz 96 system with noise.