How to make sense of this calculus notation, Advanced College Level I have $f(x)$=$(2x,e^x)$
what does this notation mean? Notation: $Df(\frac{∂}{∂x})$
Certainly $Df(x)$=$(2,e^x)$
but how can I replace $x$ with $\frac{∂}{∂x}$?
Particularly, how can I make sense of $e^{\frac{∂}{∂x}}$
 A: In your case $f$ is a map from $\mathbb{R}^1$ to $\mathbb{R}^2$, so at any point $x\in \mathbb{R}^1$ $Df$ is a linear map from $T\mathbb{R}^1_x$ to $T\mathbb{R}^2_{f(x)}$, and is given by the $1\times 2$ matrix $(2,e^x)^T$. To clarify the notation, $\frac{\partial}{\partial x}$ is notation for the unit tangent vector in the tangent space at $x$ (pointing in the positive direction), so if $u$ and $v$ are your coordinates on $\mathbb{R}^2$ we have $Df\left( \frac{\partial}{\partial x}\right)=2\frac{\partial}{\partial u}+e^x\frac{\partial}{\partial v}$ (where $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ are the unit tangent vectors in the positive $u$ and $v$ directions, which are a basis for the tangent space at $f(x)$).
A: The keyword is Operational Calculus (let Google be your friend). 
The following reference is an introductory exposure of the subject: Re: Why exp(-st) in the Laplace Transform?
About the second part of your question. Any differentiable function can be developed into a Taylor series:
$$
   f(x+a) = f(x) + a.f'(x) + \frac{1}{2} a^2.f''(x) + \frac{1}{3!} a^3 f'''(x) + \cdots
$$
Write as follows:
$$
   f(x + a) = \left[ 1 + a\frac{d}{dx} + \frac{1}{2} a^2\frac{d^2}{dx^2} + \frac{1}{3!} a^3\frac{d^3}{dx^3} +\cdots \right] f(x) $$
$$   = \left[ 1 + \left(a\frac{d}{dx}\right) + \frac{1}{2} \left(a\frac{d}{dx}\right)^2 + \frac{1}{3!} \left(a\frac{d}{dx}\right)^3 +\cdots \right] f(x) $$
The series expansion of $e^x$ is recognized in the expression between the square brackets. Therefore
we write symbolically:
$$
   f(x+a) = e^{a\frac{d}{dx}} f(x)
$$
