First three derivatives of an inverse function ($g^{\prime }=1/f^{\prime },g^{\prime \prime }=-f^{\prime \prime }/(f^{\prime }{)}^3,g^{\prime \prime \prime }=[3(f^{\prime \prime }{)}^2-f^{\prime }{f}^{\prime \prime \prime }]/(f^{\prime }{)}^5$)

Let a three-times continuously differentiable real function $f(g)$ have $df/dg\ne 0$. The inverse function theorem gives a local inverse $g(f)$. Writing $p=df/dg$, $q=d^{2}f/d{g}^2$ and $r=d^{3}f/d{g}^3$, its derivatives areIndeed $d/df=p^{-1}d/dg$ by the chain rule. Apply this operator successively to $g$, $p^{-1}$ and $-q{p}^{-3}$. Every forward derivative on the right is evaluated at the inverse image $g(f)$.