Are there elementary functions whose derivatives can't be written in elementary form

Question

I'm probably using some terms wrong in asking this so let me know in the comments.

If I have any closed-form function, f(x), will the derivative also always be able to be written in closed form? Or do we know of functions whose derivative we can't deduce, only brute-force/analytically calculate the values of a specific point one point at a time.

Non-technical way of asking this, if I give you a pencil and paper and write down a function, will you always be able to then write down the derivative of that function using only the paper and pencil? Or can some derivative forms just not be solved (generalized?) but only approximated with computers

@Somos I might be using some wrong terms but I wouldn't consider that elementary. If the original function can't itself be generalized in closed-form, I'm not sure I'd be surprised that its derivative also can't. — Albert Renshaw, Mar 09 '20 at 03:50
@Somos: The gamma function is probably the best known example of a function that doesn't even satisfy an algebraic differential equation, which is a much, much more extensive notion. Indeed, even elliptic functions and Bessel functions satisfy algebraic differential equations, in fact, very simple ones of low order and low degree (of order $2$ and degree $3).$ — Dave L. Renfro, Mar 09 '20 at 08:29
The derivative of gamma can be expressed using the digamma function. — Somos, Mar 09 '20 at 18:29

score 0 · Answer 1 · answered Mar 09 '20 at 02:28

For all of the elementary functions, we have formulas for the derivatives of the basic functions. This, we have an explicit formula for all derivatives of elementary functions.

If you take a power series or the solution to a differential equation, it’s easy to come up with a function which is not elementary and whose derivative is less explicit.

Are there elementary functions whose derivatives can't be written in elementary form

1 Answers1