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?

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.
