# Analytical derivative of a signal $y(t)$ wrt to a signal $x(t)$

I am running a sensitivity study on the model $$y(t) = x(t - \tau)$$ where $$y(t)$$ and $$x(t)$$ are 2 time signals and $$\tau$$ a time lag. Basically I want to study the sensitivity of $$y$$ to a change in $$x$$. Note that the model is very simple just for the sake of illustrating my question.

What I think about is the derivative of $$y$$ with respect to $$x$$, but I cannot see clearly how this could be done. I know the result is 1, but the way I see it is:

$$$$\frac{\partial y}{\partial x(t-\tau)}(t) = \frac{\partial y}{\partial t}(t)\frac{\partial t}{\partial x(t - \tau)} = \frac{\partial x}{\partial t}(t-\tau) \frac{\partial t}{\partial x(t-\tau)} = 1$$$$ so that a change in $$x$$ at $$t-\tau$$ is seen by the same change at a future time $$t$$ in $$y$$.

Is that a rigorous and mathematical way to look at it? I know that I might have messed up the equation and the time arguments, but I would like to learn the rigorous way to write such a problems.

Maybe to show the general case, I am interested in the following analytical models: $$y(t) = \sum_i f_i^{p_i}(x_i(t - \tau_i))$$ where $$f_i$$ is an elementary function. For instance if $$f_i(x) = x \ \forall i$$, then $$y(t) = \sum_i x_i^{p_i}(t - \tau_i)$$. What i am interested in is sensitivities of $$y$$ wrt to the different $$x_i$$'s.

Given for example $$y(t) = x(t) + z(t-\tau_1)k(t)$$, where $$x(t), y(t), z(t), k(t)$$ time signals, I can write $$y(t) = F(t, x(t), z(t-\tau_1), k(t))$$ for some function $$F$$. I want to study the following:
For a fixed time $$t$$ \begin{align} & \lim_{\epsilon \to 0} \frac{F(t, x(t) + \epsilon, z(t-\tau_1), k(t)) - F(t, x(t),z(t-\tau_1), k(t))}{\epsilon} = \\ &\lim_{\epsilon \to 0} \frac{x(t) + \epsilon + z(t-\tau_1)k(t) - x(t) - z(t-\tau_1)k(t)}{\epsilon} = \lim_{\epsilon\to 0} \frac{\epsilon}{\epsilon} = 1. \end{align} This quantity is what I refer to $$\frac{\partial y}{\partial x(t)}(t)$$. For example also, \begin{align} \frac{\partial y}{\partial z(t-\tau_1)}(t) := &\lim_{\epsilon \to 0} \frac{F(t, x(t), z(t-\tau_1) + \epsilon, k(t)) - F(t, x(t),z(t-\tau_1), k(t))}{\epsilon} \\ & \lim_{\epsilon \to 0} \frac{(z(t-\tau_1)+\epsilon)k(t) - z(t-\tau_1)k(t)}{\epsilon} = k(t) \end{align}