# Taylor expansion of Gaussian process function with input noise

I am reading "Gaussian Process Training with Input Noise" by Andrew McHutchon and Carl Edward Rasmussen, where it is assumed that the inputs $$x$$ are noisy measurements of the actual latent input $$\tilde{x}$$ , i.e $$x = \tilde{x}+\epsilon_x$$ with $$\epsilon_x\sim N(0,\Sigma_x)$$. The observation is: $$y=f(\tilde{x}+\epsilon_x)+\epsilon_y$$ under a GP model $$f$$.

The authors consider taylor expansion of the above expression up to the first order terms as follows: $$y = f(x) + \epsilon_x^T\delta_\bar{f}+\epsilon_y$$, where $$\delta_\bar{f}$$ represents the derivative of the mean of the GP function. I`d like to know why this 1st order expansion would be an accurate-enough approximation for the input error propagation.

If anyone could help, I would be very grateful.