Is the following analysis correct? Performing integral to get unknown functions

Question

I have two unknown, complex functions $u_{k}(x)e^{i(2\pi l/a)x},v_{k'}(x)e^{-i(2\pi l'/a)x}$ with $k,k',l,l'\in \mathbb{Z}$ and an expression involving both as: \begin{eqnarray} \int_{-\infty}^{+\infty}dx.e^{i(2\pi l/a)x}u_{k}(x)\left(i\frac{dv_{k'}(x)e^{-i(2\pi l/a)x}}{dx}\right)=\left(\frac{2\pi k}{N.a} + \frac{2\pi l'}{a} \right)\delta_{kk'}\delta_{ll'} \end{eqnarray} where $a,N$ are constants. My goal is to come up with a way to compute these functions, in order to match both sides, or a condition that both need to be satisfied in order to later try an ansatz. I am splitting the integral as an infinite sum: \begin{eqnarray} \int_{-\infty}^{+\infty}dx \to \sum_{s}\int_{s.a}^{(s+1).a}dx \end{eqnarray} and defining a new variable $z=x-a.s$, I rewrite the integral as \begin{eqnarray} \sum_{s}\int_{0}^{a}dz e^{i2\pi z(l-l')/a}\left[ \lambda +\frac{2\pi l'}{a} \right]u_{k}(z+sa)v_{k'}(z+sa) && \end{eqnarray} where I have used inside the integral a condition that can potentially match the rhs: \begin{eqnarray} i\frac{d(v_{k'}(x)e^{i2\pi l'x/a})}{dx} \to i\frac{d( v_{k'}(z+sa)e^{-i2\pi z l'/a})}{dz}=\lambda v_{k'}(z+sa) \end{eqnarray} from where I can extract $\lambda$ and see that the rhs is satisfied if the functions satisfy: \begin{eqnarray} \sum_{s}u_{k}(z+sa)v_{k'}(z+sa)=\delta_{kk'} \end{eqnarray} The question concerns the derivative operator, and whether is correct to assume that: \begin{eqnarray} \frac{d}{dx} = \frac{d}{dz} \end{eqnarray} Are the above manipulations correct?