# how to understand sensitivity of a system

I am reading a book about measurement. It introduces a concept of sensitivity of the system: the output of the system when the input undergoes a small change. Assuming the system can be described with a function $f(x)$ with $x$ is the input, According to the definition, I think the sensitivity is given by

$$S = \frac{f(x+\Delta x)}{\Delta x}$$

I think it is the derivative, isn't it? If it is derivative, it could be positive, negative or zero. So my question is what is the physical significance of having a negative sensitivity?

My second question is if the system is nonlinear, so $f(x)$ is a nonlinear function, so does the above definition of sensitivity still work? Thanks.

$$S = \lim_{\Delta x \to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}=\dfrac{df}{dx}$$
So if you plot $f(x)$ against $x$ the sensitivity is the gradient.
It tells you how much the output $f(x)$ changes per unit change in input $x$.
It works for a non-linear function but only for small changes in $x$ about a chosen value of $x$ in the same way as incremental resistance is used.
A negative sensitivity means that as the input $x$ increases so the output $f(x)$ decreases.