# Differentiating the equation of curvature equation

#### suren

Hi everyone,

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

#### idontknow

Have you found the derivative of (1) yet ?

#### idontknow

Is this what you are searching for ?
$$\displaystyle k'=0=\dfrac{y'''(1+y'^2 )^{3/2}-3\sqrt{1+y'^2 } y'y''^2 }{1+y'^3 }$$.

$$\displaystyle y'''(1+y'^2 )^{3/2}-3\sqrt{1+y'^2 } y'y''^2=0$$.

#### suren

i do not know what the correct answer should be, but i believe this is correct. Are you able to differentiate this equation again, i.e find k'' ?. Because i require an equation with y'''' in it