Functions with no closed-form derivative

Question

There are many well-known functions (such as $x^x$) which have no closed-form integral. However, are there any elementary functions whose derivatives cannot be expressed in a closed-form manner?

Answer 1

If $f(x)$ is built out of the "usual" functions (rational functions, exponentials, logarithms, trig, inverse trig) using addition, subtraction, multiplication, division, or composition, then $f'(x)$ is also.

The proof is on induction on complexity. For the base case, all of "easy" functions work: If $f(x) = x$ or $e^x$ or $\ln x$ or $\sin x$ or $\operatorname{trig}(x)$ or $\operatorname{arctrig}(x)$ (where "trig" stands for one of the 6 trig functions), then $f'(x)$ is elementary.

Next, suppose $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$ are all elementary.

Is $f(x) + g(x)$ also elementary? Yes because $(f(x)+g(x))' = f'(x) + g'(x)$ is a sum of elementary functions, hence is elementary. The same argument works for subtraction.

For multiplication and division, use the product/quotient rules. For composition, use the chain rule.

Then, e.g., composition works as follow: $f(g(x))' = f'(g(x))g'(x)$. But $f'(g(x))$ is elementary because $f'$ and $g$ are and elementary things are closed under composition. Further, $g'(x)$ is elementary by assumption. So, $f'(g(x))g'(x)$ is a product of elementary functions, hence elementary.

Answer 2

The elementary functions are generated by applying finite numbers of algebraic functions (algebraic operations), $\exp$ and/or $\ln$. The derivatives of all of these functions are in closed form, and chain rule gives derivatives in closed form. All algebraic functions can be differentiated by applying sum rule, product rule and chain rule. Therefore the function expressions of the elementary functions are closed under differentiation. That means the derivative of each differentiable elementary function has a representation in closed form.

The same holds for repeated differentiation (higher derivatives). There are higher sum rule, higher product rule (General Leibniz rule) and higher chain rule (Faà di Bruno's formula).