Scaling function in wavelet analysis

Question

Prove that for a scaling function $\phi$, we have:

$$\int_\Bbb R\phi_k^{'}(x) \ \phi_l(x) \ dx\;=\;\int_\Bbb R \phi_k(x) \ \phi_l^{'}(x) \ dx$$ where $\phi_l^{'}(x)$ and $\phi_k^{'}(x)$ denotes the derivatives of translated scaling function $\phi$.

Answer 1

I have tried out by doing integration by parts. Consider $\int_\mathbb{R} \phi_k^{'}(x) \ \phi_l(x) \ dx=\phi_l(x) \ \phi_k(x)\bigg|_{-\infty}^{\infty}-\int_\mathbb{R} \phi_l^{'}(x) \ \phi_k(x) \ dx.$

Now, in the above equation, I want to know how one can prove that $\phi_l(x) \ \phi_k(x)\bigg|_{-\infty}^{\infty}=0$. One reason which I think must be that $\phi(x)$ takes finite value at infinity.