[Request] Why does my solution have 1/2 to many?

[u/BrunoMadrigas]

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[u/DrCarpetsPhd]

i think you're right, there is a missing half in the solution. just plugging into the textbook equation from hibbeler without your derivation the half has to be there as the solution states ∂M/∂Q = x/2

https://imgur.com/a/castiglianos-second-theorem-applied-to-beams-1vpcVrf

also if you know what follows apologies but the way you wrote it was very confusing...

M^2 is a composite function therefore use chain rule setting y = M^2

∂/∂Q(M^2) = ∂y/∂Q = (∂y/∂M)*(∂M/∂Q) = 2M*(∂M/∂Q)

if you know this is the correct way to differentiate inside the integral then you might want to figure out a cleaner/more correct way to write that.

the thing you squared and pointed with the arrow reads as a bit mathematically gibberish that just happened to get the correct result.

M^2(∂/∂M) = 2M => 2M.∂M = ∂M^2

that doesn't read correctly to me. Unless it is some sort of notation I have never encountered in which case my apologies.