How obtain the n-th derivative of a function with an implicit variable?

I'm interested in obtain the analytical n-th derivative with respect to the variable $x$ of an implicit function like $V(h(x))$ where $h(x)$ is an implicit equation and can not be obtained the variable $h$ in terms of $x$. But we know that $\frac{dh}{dx}=f(h)$, so that we can make the derivatives with respect to $h$ using the chain rule:

$\frac{dV(h(x))}{dx}=\frac{dV(h)}{dh}\frac{dh}{dx}=\frac{dV(h)}{dh}\cdot f(h)$

$\frac{d^2V(h(x))}{dx^2}=\frac{d^2V(h)}{dh^2} \left(\frac{dh}{dx}\right)^2+\frac{dV(h)}{dh}\frac{df(h)}{dh}f(h) =\frac{d^2V(h)}{dh^2} f(h)^2+\frac{dV(h)}{dh}\frac{df(h)}{dh}f(h) $

. . .

And here is the problem, how can be obtained this in Mathematica?, since using the command D[V[h[x]], {x, n}] it doesn't work.

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Category: calculus and analysis Time: 2016-07-30 Views: 13

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