Suppose we have define function $k:$
$$k(x):=h\circ g^{-1}(x)$$
If $g^{-1}:x \rightarrow y$ such that $g^{-1}(x)=y$, it seems plausible that $k(x)= h(y)$. However, is it true that $$\frac{d^2h(y)}{d^2y}=\frac{d^2k(x)}{d^2x}?$$
On the one hand I think that $\frac{d^2(h\circ g^{-1}(x))}{d^2(x)} \neq \frac{d^2h(y)}{d^2y}$. However, if $k(x)= h(y)$, it's hard to see how the second derivatives may be different.