How can I determine a general formula for the nth derivative of any continuous function $f(x)$ differentiable at least $n$ times?

I know how to do it with easier functions, but is there a universal method which can be applied to all continuous functions differentiable at least $n$ times(introduced to in a second year calculus class)?

I can do it for easy ones like $\sin(x)$ and $\cos(x)$, but $\sin(x^2)$ and $\tan(x)$ and $\ln(1+x^2)$ are proving to be very difficult.

Thanks.

The functions that you list, $\sin(x^2)$ and $\tan(x)$ and $\ln(1+x^2)$ are indeed infinitely many times differentiable, you just need to learn differentiation rules, e.g. product, quotient, chain, power rule, etc, and differentiate these functions as many times as you wish (and, perhaps, in some cases find a general formula for the $n$th derivative). Well, for some functions it will indeed be difficult to find a general formula. You may take the first few derivatives, and try to guess the formula for the $n$th derivative in each specific case. If you guess is correct, then you may prove it by induction.

There are four parts in this answer.

Part I

The Faa di Bruno formula (see Theorem 11.4 in the book [1] and Theorem C on page 139 in the monograph [2] below) can be described in terms of the Bell polynomials of the second kind $B_{n,k}\bigl(x_1,x_2,\dotsc,x_{n-k+1}\bigr)$ by \begin{equation}\label{Bruno-Bell-Polynomial}\tag{1} \frac{\textrm{d}^n}{\textrm{d}x^n}f\circ h(x)=\sum_{k=0}^nf^{(k)}(h(x)) B_{n,k}\bigl(h'(x),h''(x),\dotsc,h^{(n-k+1)}(x)\bigr), \quad n\ge0. \end{equation} In Theorem 5.1 of the paper [3] and Section 3 of the paper [4] below, it was found that \begin{equation}\label{Bell-x-1-0-eq}\tag{2} B_{n,k}(x,1,0,\dotsc,0) =\frac{1}{2^{n-k}}\frac{n!}{k!}\binom{k}{n-k}x^{2k-n}, \quad n\ge k\ge0, \end{equation} where $\binom{0}{0}=1$ and $\binom{p}{q}=0$ for $q>p\ge0$. Consequently, as an example, we have \begin{equation*} \Bigl(e^{x^2}\Bigr)^{(n)} =e^{x^2}\frac{n!}{(2x)^n} \sum_{k=0}^{n}\binom{k}{n-k}\frac{(2x)^{2k}}{k!}, \quad n\ge0. \end{equation*} Similarly, by virtue of the formulas \eqref{Bruno-Bell-Polynomial} and \eqref{Bell-x-1-0-eq}, general formulas for the $n$th derivatives of the functions $\sin\bigl(x^2\bigr)$ and $\ln\bigl(1+x^2\bigr)$ can be derived readily as \begin{equation}\tag{+} \bigl[\sin\bigl(x^2\bigr)\bigr]^{(n)} =\frac{n!}{(2x)^n}\sum_{k=1}^n\binom{k}{n-k}\frac{(2x)^{2k}}{k!} \sin\biggl(x^2+\frac{k\pi}{2}\biggr), \quad n\ge0 \end{equation} and \begin{equation}\tag{#} \bigl[\ln\bigl(1+x^2\bigr)\bigr]^{(n)} =\frac{n!}{(2x)^n}\sum_{k=1}^n\binom{k}{n-k}\frac{(-1)^{k-1}}{k}\frac{(2x)^{2k}}{(1+x^2)^{k}}, \quad n\ge1. \end{equation}

Part II

In Theorem 1.2 of the paper [5] below, it was derived that \begin{multline}\label{bell-sin-eq}\tag{3} B_{n,k}\biggl(-\sin x,-\cos x,\sin x,\cos x,\dotsc, \cos\biggl[x+\frac{(n-k+1)\pi}{2}\biggr]\biggr)\\ =\frac{(-1)^k\cos^kx}{k!}\sum_{\ell=0}^k\binom{k}{\ell}\frac{(-1)^\ell}{(2\cos x)^\ell} \sum_{q=0}^\ell\binom{\ell}{q}(2q-\ell)^n \cos\biggl[(2q-\ell)x+\frac{n\pi}2\biggr] \end{multline} and \begin{multline}\label{bell-sin=ans}\tag{4} B_{n,k}\biggl(\cos x,-\sin x,-\cos x,\sin x,\dotsc, \sin\biggl[x+\frac{(n-k+1)\pi}{2}\biggr]\biggr)\\ =\frac{(-1)^k\sin^{k}x}{k!}\sum_{\ell=0}^k\binom{k}{\ell}\frac1{(2\sin x)^{\ell}} \sum_{q=0}^\ell(-1)^q\binom{\ell}{q}(2q-\ell)^n \cos\biggl[(2q-\ell)x+\frac{(n-\ell)\pi}2\biggr] \end{multline} for $n\ge k\ge0$. Since $\sin\bigl(x\pm\frac{\pi}{2}\bigr)=\pm\cos x$ and $\cos\bigl(x\pm\frac{\pi}{2}\bigr)=\mp\sin x$, the formulas \eqref{bell-sin-eq} and \eqref{bell-sin=ans} are equivalent to each other. These two formulas \eqref{bell-sin-eq} and \eqref{bell-sin=ans} can be applied to establish general formulas of the $n$th derivatives for functions of the types $f(\sin x)$ and $f(\cos x)$, such as $\sin^\alpha x$, $\cos^\alpha x$, $\sec^\alpha x$, $\csc^\alpha x$, $e^{\pm\sin x}$, $e^{\pm\cos x}$, $\ln\cos x$, $\ln\sin x$, $\ln\sec x$, $\ln\csc x$, $\sin\sin x$, $\cos\sin x$, $\sin\cos x$, $\cos\cos x$, $\tan x=-(\ln\cos x)'$, and $\cot x=(\ln \sin x)'$, if the general formula for the $n$th derivative of $f$ is computable.

Part III

Define the falling factorial of $\alpha\in\mathbb{C}$ by \begin{equation*}%\label{Fall-Factorial-Dfn-Eq} \langle\alpha\rangle_n= \prod_{k=0}^{n-1}(\alpha-k)= \begin{cases} \alpha(\alpha-1)\dotsm(\alpha-n+1), & n\ge1;\\ 1,& n=0. \end{cases} \end{equation*} In Remark 3.1 on page 88 in the paper [17] below, the formula \begin{equation}\label{Bell-1-lambda}\tag{5} B_{n,k}\Biggl(1, 1-\lambda, (1-\lambda)(1-2\lambda),\dotsc, \prod_{\ell=0}^{n-k}(1-\ell\lambda)\Biggr) =\frac{(-1)^k}{k!} \sum_{\ell=0}^k (-1)^{\ell} \binom{k}{\ell} \prod_{q=0}^{n-1}(\ell-q\lambda) \end{equation} for $\lambda\in\mathbb{C}$ and $n\ge k\ge0$ was concluded. In Theorem 2.1 on page 165 in the paper [11] below, it was discovered that \begin{equation}\label{Bell-fall-Eq}\tag{6} B_{n,k}(\langle\alpha\rangle_1, \langle\alpha\rangle_2, \dotsc,\langle\alpha\rangle_{n-k+1}) =\frac{(-1)^k}{k!}\sum_{\ell=0}^{k}(-1)^{\ell}\binom{k}{\ell}\langle\alpha\ell\rangle_n \end{equation} for $\alpha\in\mathbb{C}$ and $n\ge k\ge0$. These two formulas \eqref{Bell-1-lambda} and \eqref{Bell-fall-Eq} are equivalent to each other.

By virtue of the Faa di Bruno formula \eqref{Bruno-Bell-Polynomial} and any one of the formulas \eqref{Bell-1-lambda} and \eqref{Bell-fall-Eq}, one can establish general formulas of the $n$th derivatives for functions of the type $f(x^\alpha)$ for $\alpha\in\mathbb{R}$, such as $e^{x^\alpha}$, $\sin[(a+bx)^\alpha]$, and $\ln(1\pm x^\alpha)$, if the $n$th derivative of the function $f$ is computable.

Part IV

The $n$th derivative formulas for the tangent function $\tan x=\dfrac{\sin x}{\cos x}$ and the cotangent function $\cot x=\dfrac{\cos x}{\sin x}$ can also be computed by the formula \begin{equation}\label{Sitnik-Bourbaki}\tag{7} \frac{\textrm{d}^k}{\textrm{d}z^k}\biggl(\frac{u}{v}\biggr) =\frac{(-1)^k}{v^{k+1}} \begin{vmatrix} u & v & 0 & \dotsm & 0\\ u' & v' & v & \dotsm & 0\\ u'' & v'' & 2v' & \dotsm & 0\\ \dotsm & \dotsm & \dotsm & \ddots & \dotsm\\ u^{(k-1)} & v^{(k-1)} & \binom{k-1}1v^{(k-2)} & \dots & v\\ u^{(k)} & v^{(k)} & \binom{k}1v^{(k-1)} & \dots & \binom{k}{k-1}v' \end{vmatrix}. \end{equation} For details on the formula \eqref{Sitnik-Bourbaki}, please refer to https://math.stackexchange.com/a/4261705/945479.

The texts in the first three parts above are extracted and modified from Sections 1.3, 1.4, and 1.6 in the paper [6] below.

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