complex analysis – Understanding proof of analyticity of $overline{fleft (overline zright )}$

I’m reading Marsden and Hoffman’s book of Complex Analysis, and I’m struggling to understand the following example

Let $A$ be an open subset of $mathbb{C}$ and $A^*={z|overline zin A}$. Suppose $f$ is analytic on $A$, and define a function $g$ on $A^*$ by $g(z)=overline{fleft (overline zright )}$. Show that $g$ is analytic on $A^*$.

The proof provided by the book is

If $f(z)=u(x,y)+iv(x,y)$, then $g(z)=overline{fleft (overline zright )}=u(x,-y)-iv(x,-y)$.
We check the Cauchy-Riemann equations for $g$ as follows:$$dfrac{partial}{partial x}(operatorname{Re}g)=dfrac{partial}{partial x}u(x,-y)=left .dfrac{partial u}{partial x}right |_{(x,-y)}=left .dfrac{partial v}{partial y}right |_{(x,-y)}=dfrac{partial}{partial y}(-v(x,-y))=dfrac{partial}{partial y}(operatorname{Im}g).$$

I’m failing to understand why $left .dfrac{partial v}{partial y}right |_{(x,-y)}=dfrac{partial}{partial y}(-v(x,-y))$.
When I use the chain rule with $h(x,y)=(x,-y)$ I get $dfrac{partial}{partial y}(v(x,-y))=dfrac{partial}{partial y}(v(h(x,y)))=left .dfrac{partial (vcirc h)}{partial y}right |_{(x,y)}$ andbegin{align*}begin{pmatrix}dfrac{partial}{partial x}(vcirc h)&dfrac{partial}{partial y}(vcirc h)end{pmatrix} & =D(vcirc h) \
& =D(v)D(h) \
& =begin{pmatrix}dfrac{partial}{partial x}v&dfrac{partial}{partial y}vend{pmatrix}begin{pmatrix}dfrac{partial}{partial x}h_1&dfrac{partial}{partial y}h_1\dfrac{partial}{partial x}h_2&dfrac{partial}{partial y}h_2end{pmatrix} \
& =begin{pmatrix}dfrac{partial}{partial x}v&dfrac{partial}{partial y}vend{pmatrix}begin{pmatrix}1&0\0&-1end{pmatrix} \
& =begin{pmatrix}dfrac{partial}{partial x}v&-dfrac{partial}{partial y}vend{pmatrix}.
end{align*}
Hence $dfrac{partial}{partial y}(v(x,-y))=left .dfrac{partial (vcirc h)}{partial y}right |_{(x,y)}=-left .dfrac{partial v}{partial y}right |_{(x,y)}$.
So $left .dfrac{partial v}{partial y}right |_{(x,y)}=dfrac{partial}{partial y}(-v(x,-y))=left .dfrac{partial v}{partial y}right |_{(x,-y)}$, but the last one doesn’t feel right.

Can anyone help me to prove that $left .dfrac{partial v}{partial y}right |_{(x,y)}=dfrac{partial}{partial y}(-v(x,-y))$ or at least point out my mistake in the chain rule?