Third Derivative of Inverse Function

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Theorem

Let $f$ be a real function which is of differentiability class $3$.

Let $f$ have an inverse $f^{-1}$, likewise of differentiability class $3$.

Then:

$\dfrac {\d^3 x} {\d y^3} = -\paren {\dfrac {\d^3 y} {\d x^3} \dfrac {\d y} {\d x} - 3 \paren {\dfrac {\d^2 y} {\d x^2} }^2} \paren {\dfrac {\d y} {\d x} }^{-5}$


Proof

\(\ds \dfrac {\d^3 x} {\d y^3}\) \(=\) \(\ds \map {\dfrac \d \d y } {\dfrac {\d^2 x} {\d y^2} }\) Definition of Third Derivative
\(\ds \) \(=\) \(\ds \map {\dfrac \d {\d y} } {-\dfrac {\d^2 y} {\d x^2} \paren {\dfrac {\d y} {\d x} }^{-3} }\) Derivative of Inverse Function
\(\ds \) \(=\) \(\ds -\paren {\paren {\dfrac {\d y} {\d x} }^{-3} \map {\dfrac \d {\d y} } {\dfrac {\d^2 y} {\d x^2} } + \dfrac {\d^2 y} {\d x^2} \map {\dfrac \d {\d y} } {\paren {\dfrac {\d y} {\d x} }^{-3} } }\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds -\paren {\paren {\dfrac {\d y} {\d x} }^{-3} \dfrac {\d x} {\d y} \map {\dfrac \d {\d x} } {\dfrac {\d^2 y} {\d x^2} } + \dfrac {\d^2 y} {\d x^2} \dfrac {\d x} {\d y} \map {\dfrac \d {\d x} } {\paren {\dfrac {\d y} {\d x} }^{-3} } }\) Derivative of Composite Function
\(\ds \) \(=\) \(\ds -\paren {\paren {\dfrac {\d y} {\d x} }^{-3} \dfrac {\d x} {\d y} \dfrac {\d^3 y} {\d x^3} + \dfrac {\d^2 y} {\d x^2} \dfrac {\d x} {\d y} \paren {-3 \paren {\dfrac {\d y} {\d x} }^{-4} } \map {\dfrac \d {\d x} } {\dfrac {\d y} {\d x} } }\) Definition of Third Derivative, Derivative of Composite Function
\(\ds \) \(=\) \(\ds -\paren {\paren {\dfrac {\d y} {\d x} }^{-3} \dfrac {\d x} {\d y} \dfrac {\d^3 y} {\d x^3} + \paren {\dfrac {\d^2 y} {\d x^2} }^2 \dfrac {\d x} {\d y} \paren {-3 \paren {\dfrac {\d y} {\d x} }^{-4} } }\) Definition of Second Derivative
\(\ds \) \(=\) \(\ds -\paren {\dfrac {\d^3 y} {\d x^3} - 3 \paren {\dfrac {\d^2 y} {\d x^2} }^2 \paren {\dfrac {\d y} {\d x} }^{-1} } \paren {\dfrac {\d y} {\d x} }^{-3} \dfrac {\d x} {\d y}\) simplifying
\(\ds \) \(=\) \(\ds -\paren {\dfrac {\d^3 y} {\d x^3} - 3 \paren {\dfrac {\d^2 y} {\d x^2} }^2 \paren {\dfrac {\d y} {\d x} }^{-1} } \paren {\dfrac {\d y} {\d x} }^{-3} \paren {\dfrac {\d y} {\d x} }^{-1}\) Derivative of Composite Function
\(\ds \) \(=\) \(\ds -\paren {\dfrac {\d^3 y} {\d x^3} \dfrac {\d y} {\d x} - 3 \paren {\dfrac {\d^2 y} {\d x^2} }^2} \paren {\dfrac {\d y} {\d x} }^{-5}\) simplifying and manipulating into required form

$\blacksquare$


Sources

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