 First three derivatives of an inverse function

ID: first-three-derivatives-of-an-inverse-function

f (g)

have

df / d g \neq = 0

. The inverse function theorem gives a local inverse

g (f)

p = df / d g

q = d^{2} f / d g^{2}

and

r = d^{3} f / d g^{3}

\frac{d g}{df} = \frac{1}{p}, \frac{d ^{2} g}{d f ^{2}} = - \frac{q}{p ^{3}}, \frac{d ^{3} g}{d f ^{3}} = \frac{3 q ^{2} - p r}{p ^{5}} .

Indeed

d / df = p^{- 1} d / d g

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)

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