In [this](http://conal.net/papers/beautiful-differentiation/beautiful-differentiation-long.pdf) paper, the author says that differentiation is a "metafunction" that maps one function (e.g., $x^2$) to another ($2x$). On page two, he says

> 
> For now, give differentiation the following higher order type:
> 
> 
> $d:: (a \rightarrow a) \rightarrow (a \rightarrow a)$
> 
> 
> 

So far, it makes sense to me. Then he says the sum rule (i.e., for derivatives) is short hand for:

> 
> $d(\lambda x \rightarrow u \ x + v\ x) \equiv \lambda x \rightarrow d \ u \ x \ + \ d \ v \ x $
> 
> 
> 

Where did $\lambda$ come from? What is the syntax of this thing trying to express?

The author continues by saying

> 
> These more precise formulations are tedious...Fortunately, there is an
> alternative...The trick is to add numeric overloadings for functions,
> so that numeric operations apply point-wise. For instance,
> 
> 
> $u + v \equiv \lambda x \rightarrow u \ x + \ v \ x$
> 
> 
> 

At this point, I no longer understand the expression at all. I would appreciate it if someone can explain the syntax and what the equivalence means.