Derivative of a signal $y(t)$ wrt to another signal $x(t)$

Question

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.

EDIT:

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

Answer 1

In essence, what you are asking is the ratio of how $y$ changes at $t$ to how $x$ changes at $t$. That is: $$ \frac{dy}{dx}(t) = \frac{\frac{dy}{dt}(t)}{\frac{dx}{dt}(t)}. $$ Using that $y(t)=x(t-\tau)$, you then get that this is equal to $$ \frac{dy}{dx}(t) = \frac{\frac{dx}{dt}(t-\tau)}{\frac{dx}{dt}(t)}. $$ But this is definitely not equal to one: For example, for $x(t)=t^2$, you would obtain $$ \frac{dy}{dx}(t) = \frac{\frac{dx}{dt}(t-\tau)}{\frac{dx}{dt}(t)} = \frac{t-\tau}{t}. $$ It is true, however, that $$ \lim_{\tau\rightarrow 0}\frac{dy}{dx}(t) = \lim_{\tau\rightarrow 0}\frac{\frac{dx}{dt}(t-\tau)}{\frac{dx}{dt}(t)} = 1. $$

Answer 2

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.