The $n$-th derivative of $f(x)=\cosh g(x)$

Question

If we have a function

$$f(x)=\cosh\theta(x)$$

where we'll assume that $\theta$ is a sufficiently smooth function, then we can easily calculate its first few derivatives with respect to $x$, i.e. $$f'(x)=\theta'(x)\sinh\theta(x),~f''(x)=\theta''(x)\sinh(x)+\theta'(x)^2\cosh\theta(x),$$ $$f'''(x)=\{\theta'''(x)+\theta(x)^3\}\sinh\theta(x)+3\theta'(x)\theta''(x)\cosh\theta(x),$$ $$f^{(4)}(x)=(\theta''''+6\theta'\theta''^2)\sinh\theta+(4\theta'\theta'''+3\theta''^2+\theta^4)\cosh\theta,\dots$$

and so on.

Can we give the general term in the sequence for $f^{(n)}(x)$ in terms of $\theta$ and its derivatives?

Answer 1

As mentioned by zwim, one can compute this using the General formula or a pattern for the $n$th derivatives of $e^{f(x)}$?, or, more directly,

\begin{align}\frac{\mathrm d^n}{\mathrm dx^n}\cosh(g(x))&=\cosh(g(x))\sum_{k=1}^{\lfloor n/2\rfloor}B_{n,2k}\big(g'(x),\dots,g^{(n-2k+1)}(x)\big)\\&+\sinh(g(x))\sum_{k=0}^{\lfloor(n-1)/2\rfloor}B_{n,2k+1}\big(g'(x),\dots,g^{(n-2k)}(x)\big)\end{align}

using Faà di Bruno's formula.