The curvature of a line is given exactly by:
\[ \kappa = \frac{d \phi}{ds} \]
For small \(\phi\) we note that \(ds \approx dx\), therefore:
\[ \kappa \approx \frac{d \phi}{dx} \]
\[ \tan \phi = \frac{dv}{dx} \]
For small deflections, \(\tan \phi \approx \phi\), therefore:
\[ \phi \approx \frac{dv}{dx} \]