Say I have this equation:
$x(s) = \frac{H}{w} \ln\left [ \frac{V+ws}{H} + \sqrt{1+\left ( \frac{V+ws}{H} \right )^2} \right ] - \frac{H}{w} \ln\left [ \frac{V}{H} + \sqrt{1+\left ( \frac{V}{H} \right )^2} \right ] $
Since there is no closed-form solution for $H$, the approach I use to find $H$ when all other values are known is to solve it iteratively using root finder. Unfortunately, this also implies the Jacobian is estimated using finite difference. Preferably, I'd like to calculate the Jacobian analytically.
Does anyone know of an approach to calculate the derivates of $dH/dx(s)$ analytically when a closed-form solution is not available? Better yet, is there a methodology that can be applied to calculate derivative on a class of equations where there is no closed-form solution?
If the closed form of $x(s)$ is known, you could try to calculate the inverse relation of $x$ in closed form, that means its single branches (the partial inverse functions).
Replace $s$ on the right-hand side of your equation by the terms for $s(x)$ then.
Apply implicit differentiation then.