Assumption that $\delta q'$ is small in the derivation of Euler-Lagrange equations.

Hint: the basic assumption is that the Lagrangian $L(q(x),q'(x),x)$ is differentiable at $y:=(q(x),q'(x),x)$ and the variation $\delta L$ expresses the differentiability, in a suitable limit.

In other words, we are considering the increment ( $h$ is just a vector in $\mathbb R^3$, for all $x$, but the components $\delta q$ and $\delta q'$ are functions of $x$)

$$h:=(\delta q,\delta q',0)$$

and looking at

$$L(y+h)-L(y)=\langle \nabla L(y),h\rangle+O(\|h\|);~~(*)$$

the gradient $\nabla L(y)$ of $L$ at $y$ is

$$\nabla L(y)=\left(\frac{\partial L}{\partial q},\frac{\partial L}{\partial q'},\frac{\partial L}{\partial x}\right) $$

and, by definition,

$$y+h=(q(x)+\delta q,q'(x)+\delta q',x).$$

• Remark: boundary conditions

In variational problems, the vector $h$ is not completely arbitrary: in fact, if we consider small variations _(i.e. $\delta q$) of the function $q:x\mapsto q(x)$ we need to impose boundary conditions on the components of $h$: such conditions are depending on the specific variational problem under exam. They are motivated by the variational approach and usually "kill" unfriendly boundary terms (at least in classical variational problems: in quantum field theory one can accept and subsequently work with boundary terms).

For example, in presence of the variational problem

$$S[q]:=\int_a^b L(q(x),q'(x),x)dx $$

we would be interested in all small variations $\delta q$ of $q=q(x)$ s.t. $\delta q(a)=\delta q(b)=0$.

If we expand $(*)$ using the definition of gradient and scalar product, we arrive at

$$L(q(x)+\delta q,q'(x)+\delta q',x)-L(q(x),q'(x),x)=\frac{\partial L}{\partial q}\delta q+\frac{\partial L}{\partial q'}\delta q'+O(\|h\|). $$

• Remark: parameters

  In some books and papers it is customary to define the small variations $\delta q$ as

$$\delta q:=\epsilon\varphi, $$ $$\delta q':=\epsilon\varphi' $$

where $\varphi$ is any function satisfying suitable boundary conditions, and $\epsilon$ is a parameter. In this setting we are considering the increment

$$h=(\epsilon\varphi,\epsilon\varphi',0)$$

around $y=(q(x),q'(x),x)$ and the first variation $\delta L$ of $L$ is defined as

$$\delta L:=\frac{dL}{d\epsilon}|_{\epsilon=0}:=\lim_{\epsilon\rightarrow 0} \frac{L(q(x)+\epsilon\varphi,q'(x)+\epsilon\varphi',x)-L(q(x),q'(x),x)-\epsilon\left(\frac{\partial L}{\partial q}\varphi+\frac{\partial L}{\partial q'}\varphi'\right)+O(\epsilon^2)}{\epsilon}. $$

I personally find this notation quite good.