Partial derivative notation: $\left.\frac{\partial \cdot}{\partial\cdot} \right|_{u=T}$

Let $\displaystyle \ \ B(t,T):=\int_t^T f(t,s)ds$, where $f(.,.)$ is a stochastic process whose solution we don't know.

My lecture slides make the claim that:

$$f(t,T) = \frac{\partial B(t,u)}{\partial u} \Bigg|_{u=T}$$

My first question is, what's the notation $\frac{\partial .}{\partial.} \Bigg|_{u=T}$?

My second question is what is this notation?

$$\frac{\partial B(t,T)}{\partial T} \Bigg|_{T\searrow t}$$

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The first is the value of the derivative in $u=T$ (just substitution).
The second is maybe the limit where $T$ tends monotonously to $t$.
For the first question, why couldn't he have done $\frac{\partial B(t,T)}{\partial T}$, instead of using $u$ and then substituting in $T$ afterwards? –  Jase Nov 22 '12 at 17:59
The short answer is: because $T$ is the function argument, that is, considered constant in the RHS; but you can only differentiate w.r.t. independent variable. For example, put you want $f(t,5)$. The RHS becomes meaningless. –  starteleport Nov 23 '12 at 6:47