The equation for the curvature(k) is the following:

(1) k = (y'')/((1+(y')^2)^3/2)

For small deflections, this is just simplified to y'' (the numerator)

In beam theory the following equation is used:

(2) EIy'' + Py = 0, where the y'' is referring to the simplified curvature equation. If we use the proper equation, it will be

(3) EIk + Py = 0, where k is the equation above.

There is also another equation used that is the 2nd differential of equation (2):

(4) EIy'''' + Py'' = 0

I would like to know the second differential of equation (3), using k. Basically how to differentiate the equation (1) twice and plug it into equation (3)

Many Thanks