Differential structures of sum bundle, tangent bundle, dual bundle, tensor bundle, quotient bundle, etc As the title suggests, I was wondering what structure a certain vector bundle $E$ should induce in its "derived" bundles such as the sum bundle, the dual bundle, the tensor bundle, the tangent/cotangent bundle, the quotient bundle, etc. We have said during lessons that given a vector bundle $E$ there is a way to define a smooth structure on all the bundles above. At the moment I didn't think about that and thought it could have been trivial, but now I can't think of anything. I reckon there should be only one "reasonable" induced structure, but I really can't see it (actually for the sum bundle I can, but it is the most trivial). 
I was searching for someone giving me an idea, or a reference that explains this (even briefly)
 A: The standard reference I like to recommend John M. Lee, Introduction to smooth manifolds as it is very detailed and easy to read.
Lee states 'vector bundle chart lemma' which will possibly help you.
But you can have a general view on this (in general for fibre bundles, but I will restrict to the vector bundle case).
So fix some notation: Let $M$ denote a (smooth) manifold of dimension $n$, $E$ a vector bundle over $M$ of rank $k$, whith projection $\pi: E \rightarrow M$ and fix an open cover $M= \bigcup_{i \in I} U_i$ with local trivializations $\Psi_i\colon \pi^{-1}(U_i) \rightarrow U_i \times \mathbb{R}^k$.
Let $p \in U_i$ be a point and denote the fibre as $E_p:=\pi^{-1}\{p\}$. Then $E_p$ is a vector space of dimension $k$ and the trivialization $\Psi_i$ sends any vector $v_p \in E_p$ to the pair $(p, \psi_p (v_p))$, where $\psi_p \in \text{Hom}_{\mathbb{R}}(E_p, \mathbb{R}^k)$ and this assignement is smooth in $p$ in the following senese: After choosing bases, one can rewrite this as a map $U_i \rightarrow \text{GL}(k, \mathbb{R})$, $p\mapsto \psi_p$, which is smooth.
It follows that the transition maps of the local trivializations have the form
$ \Psi_i \Psi_j ^{-1}: U_i \cap U_j \times \mathbb{R}^k \rightarrow U_i \cap U_j \times \mathbb{R}^k, (p, v) \mapsto (p, \tau (p)v)$, where $p \mapsto \tau(p) \in \text{GL}(k,\mathbb{R})$ is smooth.
Thus any vector bundle gives a cocycle $\{\Phi_{ij}\colon U_i \cap U_j \rightarrow \text{GL}(k,\mathbb{R})\}_{i, j \in I}$, where $\{U_i\}_{i \in I}$ is an open cover of $M$, where all $U_i$ are domains of charts, and the $\Phi_{ij}$ satisfy the cocycle-condiction $\Phi_{ij}\Phi_{jk}=\Phi_{ik}$ and $\Phi_{ii}=\text{id}_{\mathbb{R}^k}$.
Conversely we want to see, that such cocycle determines the smooth structure of a rank $k$ vector bundle over $M$. So assume we are given a cocycle $\{\Phi_{ij}\colon U_i \cap U_j \rightarrow \text{GL}(k,\mathbb{R})\}_{i, j \in I}$, we want to construct a vector bundle $E \rightarrow M$.
Consider the set $\tilde{E}:=\bigcup_{i \in I}\{i\}\times U_i \times \mathbb{R}^k$ with the following equivalence relation: For any $p \in U_i \cap U_j \colon (i,p,v) \sim (j,p, \Phi_{ji}(p)v)$. Let $E:=\tilde{E}/ \sim$ be the set of equivalence classes $[i,p,v]$, with the projection $\pi\colon E\rightarrow M, [i,p,v] \mapsto p$.
The smooth structure of $E$ then is determined by the bijections $\Psi_i\colon \pi^{-1}(U_i) \rightarrow U_i \times \mathbb{R}^k, [i,p,v] \mapsto (p,v)$
Checking all conditions for $E$ beeing a vector bundle is now straightforward.
Now to answer your question: Fix a mainifold $M$ and consider two vector bundles $E$ and $F$ of rank $k$ and $l$. After passing to a refinement of the open covers of $M$, they are given by cocycles $\{\Phi^E_{ij}\colon U_i \cap U_j \rightarrow \text{GL}(k,\mathbb{R})\}_{i, j \in I}$ and $\{\Phi^F_{ij}\colon U_i \cap U_j \rightarrow \text{GL}(l,\mathbb{R})\}_{i, j \in I}$. Then the direct sum $E \oplus F$ is determined by the cocycle$\{\Phi^E_{ij} \oplus \Phi^F_{ij}\colon U_i \cap U_j \rightarrow \text{GL}(k+l,\mathbb{R})\}_{i, j \in I}$.
The dual bundle of $E$, denoted $E^*$, is determined by $\{U_i \cap U_j \rightarrow \text{GL}(k,\mathbb{R}), p \mapsto ((\Phi^E_{ij} (p))^T)^{-1}\}$, where $T$ denotes the transpose.
Many other constructions from linear algebra now carry over onto vector bundles and it is a good excercise to find their cocycle descritptions.
A special case is the tangent bundle. Choose an atlas $\{\Psi_i,U_i\}_{i \in I}$ of $M$ and denote $J(\Psi_i \Psi_j^{-1})$ the Jacobian of the transition functions.
The tangent bundle $TM$ now is determined by the cocycle $\{U_i \cap U_j \rightarrow \text{GL}(n,\mathbb{R}), p \mapsto J(\Psi_i \Psi_j^{-1}) \Psi_j(p)\}_{i,j \in I}$
