Derivative self-adjoint operator

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1)I have the symbol $\circ$ ?

In $-(\mathcal{L} \psi)\circ \psi^{-1}$

How to read it in math in this case ?

2)The derivative of $q(y)=-(\mathcal{L} \psi(x)) $ is $q^\prime(y)= \frac{-{[\mathcal{L}\psi]}^\prime(x)]} {\psi^\prime (x)}$

I am trying to get but I can not ? Can you help to obtain the $q^\prime(y)= \frac{-{[\mathcal{L}\psi]}^\prime(x)]} {\psi^\prime (x)}$, please ?

where $y=\psi(x) =\int_0^x 1/\sqrt{(a(z)}dz$, $x\in (0,\infty)$ and $\mathcal{L}$ is the non-positive self-adjoint operator. Thanks.