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$f(x) = \frac{1}{\cos x}$

$f'(x) = \frac{\sin(x)}{\cos^2(x)}$

$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$

$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$

$\vdots$

$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$

Some of these are easy: http://darkwing.uoregon.edu/~jcomes/251exn.pdf Others are not. Why?

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I would recommend editing the title to $f(x) = \frac{1}{\cos x}$ then $f^{(n)}(x)$, and $f^n(x)$ generally denotes $f(f(f(\dots f(f(x))\dots)))$. –  Brandon Carter Nov 2 '10 at 17:54
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4 Answers 4

up vote 14 down vote accepted

This is asking for the $n$-th derivative of the secant function. As the derivative of $\sec$ is $\sec\tan$ and that of $\tan$ is $\sec^2=1+\tan^2$ then the $n$-th derivative of $\sec$ is $\sec f_n(\tan)$ where $f_0(t)=1$ and $f_{n+1}(t)=tf_n(t)+(t^2+1)f_n'(t)$.

It's probably too much to hope to find a nice formula for the coefficients of the $f_n$. The constant coefficients of $f_{2m}$ are essentially the Euler numbers.

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For completeness: the Wolfram Functions site gives a series representation for the $n$-th derivative of the secant:

$$\frac{\mathrm d^n}{\mathrm dx^n}\sec\,x=\sum_{j=0}^\infty \frac{(-1)^j}{(2j-n)!} E_{2j} x^{2j-n}$$

where the $E_{2j}$ are the Euler numbers mentioned in Robin's answer.

There is also a finite double series representation:

$$\frac{\mathrm d^n}{\mathrm dx^n}\sec\,x=(n+1)!\sec\,x \sum _{k=0}^n \sum _{j=0}^{\left\lfloor\frac{k-1}{2}\right\rfloor} \frac{\left((-1)^k 2^{1-k} (k-2 j)^n \sec ^k x\right) \cos\left(\frac{n\pi}{2}+(k-2j)x\right)}{(k+1) j! (k-j)! (n-k)!}$$

As Qiaochu says, there's no good reason to expect that computing higher derivatives of some function is an easy task...

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That depends on what you mean by "easy." As far as the examples in the .pdf you link to, I claim that the following are true for any reasonable definition of "easy":

  • It is easy to compute the iterated derivatives of powers and logarithms.
  • It is easy to compute the iterated derivatives of solutions to homogeneous linear ODEs.
  • It is easy to compute the iterated derivatives of $x^n f(x)$ if it is easy to compute the iterated derivatives of $f(x)$.
  • It is easy to compute the iterated derivatives $f(x) + g(x)$ if it is easy to compute the iterated derivatives of $f(x)$ and of $g(x)$.

I think that covers all of them. Otherwise, there is no general reason to expect that it should be easy to compute iterated derivatives. Even at $x = 0$ they may define complicated and highly nontrivial sequences, for example the Bernoulli numbers (which are related to this problem).

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You might want to check out Faa di Bruno's formula: http://mathworld.wolfram.com/FaadiBrunosFormula.html

In your case you have $f(x) = g(h(x))$ where $g(x) = 1/x$ and $h(x) = \cos(x)$. The answer is not simple unfortunately, and involves some combinatorics.

Alternatively, notice $f(x) \cos(x) = 1$, and thus by taking the $m$th derivative of both sides we get

$$ \sum_{k=0}^m {m \choose k} f^{(k)}(x) \left( \frac{d^{m-k}}{dx^{m-k}} \cos(x) \right) = 0 $$

if $m \geq 1$. Thus, you can view the vector $(f(x),f'(x),\dots,f^{(n)}(x))$ as the solution to the system of linear equations above with $m=0,1,\dots,n$. You can invert this matrix numerically, if that if your interest. A closed form might be possible too.

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