Introduction: Up to now we were always able to find the expression of the inverse function $x(y)$ by rewriting the original function $y(x)$. Unfortunately, this is not always possible. However, even in that case we know something about this function; we are able to determine the derivative of $x(y)$ in a point without knowing the exact prescription of $x(y)$.
Definition: If $x(y)$ is the inverse of the function $y(x)$ and $y'(x)$ is the derivative of the function $y(x)$, then for the derivative $x'(y)$ of the inverse function $y(x)$ it holds that
\[x'(y) = \dfrac{1}{y'(x)} \qquad \text{where} \qquad x = x(y).\]
Definition: If $x(y)$ is the inverse of the function $y(x)$ and $y'(x)$ is the derivative of the function $y(x)$, then for the derivative $x'(y)$ of the inverse function $y(x)$ it holds that
\[x'(y) = \dfrac{1}{y'(x)} \qquad \text{where} \qquad x = x(y).\]