Is numerical approximation the only option when derivative cannot be expressed explicitly as an expression? My problem is the following: Are there any differentiable functions on $\Bbb R$ for which we don't know or can't find an explicit expression for the derivative? So is approximating the derivative numerically the only choice?
 A: If you mean that we can't find an explicit expression of the first derivative, then yes, they do exist. Take $f:\Bbb R\to \Bbb R$ as 
$$f(x)=\int_0^x \left(\int_0^t e^{\frac{-y^2}{2}}dy\right)dt.$$ Then, it can be proved that 
$$f'(x)= \int_0^xe^{\frac{-y^2}{2}}dy.$$ However, this function cannot be expressed in terms of elementary functions. That being said, you can always approximate the derivative numerically. Hope that helps.
A: There's a neat trick involving dual numbers (a number system that has a nonzero infinitesimal element $\epsilon$ such that $\epsilon^2=0$), that lets you numerically calculate derivatives of any function that you can numerically calculate in the first place, to machine precision. This procedure is no less exact than determining an analytical expression for the derivative and computing the value of that expression numerically.
A: For functions on $\Bbb R$: derivability $\iff$ differentiability. 
NOTE
That’s not the case for functions of several variables for which only one direction is true: differentiability $\implies$ derivability.
