Given function $n_{0}\left( x\right)=e^{y\left( x\right)}$ where $y\left(x\right)$ is an arbitrary function, what are the multiple derivatives of $n_{0}\left( x\right)$ in terms of the multiple derivatives of $y\left(x\right)$? For example: $\frac{dn_{0}}{dx}=e^{y}\frac{dy}{dx}$, $\frac{d^{2}n_{0}}{dx^{2}}=e^{y}\left( \left( \frac{dy}{dx}\right)^{2}+\frac{d^{2}y }{dx^{2}}\right) $, $\frac{d^{3}n_{0}}{dx^{3}}=e^{y}\left( \left( \frac{dy}{dx}\right) ^{3}+3\frac{dy}{dx}\frac{d^{2}y}{dx^{2}}+\frac{d^{3}y}{dx^{3}}\right)$. What then is a closed expression for $\frac{d^{k}n_{0}}{dx^{k}}$? Background: @DinosaurEgg solved What is the solution to $n_{i}=(e^{-x})\frac{dn_{i-1}}{dx}$ for i=0,1,2,...? that is the first half of a problem I need to solve. That solution provided me an answer in terms of the derivatives of $n_{0}\left( x\right)$, but I need it in terms of those of $y\left( x\right)$.
Asked
Active
Viewed 38 times
1
-
2The Faà di Bruno's formula gives $$ \frac{{{\rm d}^n n_0 }}{{{\rm d}x^n }} = {\rm e}^y \sum\limits_{k = 1}^n {B_{n,k} (y',y'', \ldots ,y^{(n - k + 1)} )} . $$ – Gary Nov 22 '22 at 02:29