# How to differentiate a time domain signal in the complex transfer function?

I have an input-output data set where the input is current and the output velocity. I am interested in the transfer function from current to acceleration though. So suppose: $H(s) = \frac{I(s)}{V(s)}$ where $V$ is velocity and $I$ is current. I thought that the solution would be to $H'(S) = \frac{I(s)}{sV(s)} = \frac{I(s)}{A(s)}$ where A is the laplace transform of acceleration resulting in the transfer function I'm after. Now to MATLAB: I use 'tfestimate' to obtain the first transfer function $Txy$. Now I would suppose that substituting $s$ by $j \omega$ and implementing that to obtain the magnitude as $|\frac{Txy}{j\omega}|$ would yield the correct results. But I only get the correct results by using $|Txy \cdot j\omega|$ and comparing this to the differentiated signal of $v$. What error am I making? Is it actually the complete transfer function I want to multiply with $j\omega$ to achieve the differentiation instead of the denominator? Any help would be much appreciated!

The transfer function is commonly defined as output divided by input, and not so often the other way around (as you did in your question). Also Matlab uses that more common definition. So the transfer function $H(s)$ between current $I(s)$ (input) and velocity $V(s)$ (output) would be defined by
$$H(s)=\frac{V(s)}{I(s)}\tag{1}$$
Since acceleration $A(s)$ is the derivative of velocity, we have
$$A(s)=sV(s)\tag{2}$$
$$G(s)=\frac{A(s)}{I(s)}=\frac{sV(s)}{I(s)}=sH(s)\tag{3}$$