Linear response function for a system with derivative: $U=L \frac{d I}{dt}$, expressing $U=f(I)$
Let's start with the simple system

$$y(t)=x(t)\tag{1}$$

where $y(t)$ denotes the output and $x(t)$ denotes the input. It is linear and time-invariant, and, consequently, it must be possible to represent it by the convolution integral

$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\tag{2}$$

where $h(t)$ is the system's impulse response. In this case, it turns out that $h(t)=\delta(t)$, i.e., the impulse response is a Dirac delta impulse.

Now, for a differentiator

$$y(t)=x'(t)\tag{3}$$

we "simply" (conceptually, at least) have $h(t)=\delta'(t)$, where $\delta'(t)$ is the (generalized) derivative of $\delta(t)$. So, using the concept of a generalized derivative of a distribution, we can write the input-output relationship of an ideal differentiator as the following convolution integral:

$$y(t)=\int_{-\infty}^{\infty}x(\tau)\delta'(t-\tau)d\tau\tag{4}$$