The $n^{th}$ derivative in change of variable

Question

I want to change variable $x$ in my differential equation to $t=g(x)$. I would like to create pattern for $n^{th}$ new derivative $\frac{d^ny}{dt^n}$ for any transformation depends on old derivatives $ \frac{dy}{dx}, \dots, \frac{d^ny}{dx^n} $.

Can you tell me please if the patter or algorithm exists? Thanks in advance :)

Answer 1

What you want is the Faà di Bruno's formula. This paper gives some historical background for the formula and this one presents some connections to complex analysis and combinatorics.