Let $\pi\colon E\rightarrow M$ be a smooth vector bundle.
A section $s\in\Gamma(E)\hookrightarrow\mathcal{C}^\infty(M,E)$ induces a bundle map $s^\natural\colon TM\rightarrow s^*TE$, which is defined for each point $p\in M$ by the tangent map: For $v\in T_pM$ take a curve $\gamma\colon I\rightarrow M$ with $v=\dot\gamma(0)$ and $p=\gamma(0)$, define $ds\vert_p(v):=ds\vert_{\gamma(0)}(\dot\gamma(0))=\frac{\partial}{\partial t}(s\circ\gamma(t))\vert_{t=0}$. Because $s\circ\gamma(0)=s(p)$, $ds\vert_p(v)\in T_{s(p)}E=(s^*TE)_p$. The curve $\gamma^\sharp:=s\circ\gamma$ is also called a lift of $\gamma$.
Now each fiber $E_p\subset E$ is a vectorspace and $\Gamma(E)$ is a $\mathcal{C}^\infty(M)$-Module by fiberwise addition and multiplication:
$\cdot_{\Gamma(E)}\colon\mathcal{C}^\infty(M)\times\Gamma(E)\rightarrow\Gamma(E)\colon(f,s)\mapsto f\cdot_{\Gamma(E)} s:=[p\mapsto f(p)\cdot_{E_p} s(p)]$
$+_{\Gamma(E)}\colon\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)\colon(s_1,s_2)\mapsto s_1+_{\Gamma(E)} s_2:=[p\mapsto s_1(p)+_{E_p}s_2(p)]$
I want to study the map, which takes a section to the corrisponding bundle map (essentially the tangent map of s):
$d\colon\Gamma(E)\rightarrow\Gamma(T^*M\otimes s^*TE)\colon s\mapsto ds$ (abuse of notion). This is well defined, because for every $p\in M\colon ds\vert_p$ is in $T^*_pM\otimes T_{s(p)}E$ by the computation above.
Initially I wanted to show $d$ behaves like a derivation. But this does not work, because of the following:
Although $ds\vert_p$ is a well defined element in $T_{s(p)}E$, a sum acting like "$[(ds_1\vert_p+ds_2\vert_p)(v)](f)=[ds_1\vert_p(v)](f)+[ds_2\vert_p(v)](f)$" is not a well defined tangent vector in $T_eE$, because albeit $s_1$ and $s_2$ mapping to the same fiber $E_p$, they still can still map to different points in each Fiber.
There is also a problem for fiberwise scalarmultiplication, because the multiplication is not independent of $t$. $d(f\cdot_{\Gamma(E)}s)\vert_p(v)=\frac{\partial}{\partial t}((f\cdot_{\Gamma(E)}s)\circ\gamma(t))\vert_{t=0}=\frac{\partial}{\partial t}\Big(f(\gamma(t))\cdot_{E_{\gamma(t)}}s(\gamma(t))\Big)\vert_{t=0}=?$