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:
\begin{equation}
\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
\end{equation}
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.
 A: After more thinking, I am answering my own question. What I was looking for is the following:
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}
The partial derivatives notations that I am using might make sense mathematically, but in terms of limits, that is what I was looking for.
