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I figured out the only redundancy to my problem is this:

I'll start off with a simple case, where w1,w2 are the displacements at intervals of one third along a beam.

w = 3w1/L.x (Note, x is in the numerator for all cases)

To differentiate this with respect to time, I use the chain rule as follows:

∂w/∂t = ∂w/∂x.∂x/∂t = 3w1/L.w1'

Now for the second function, which depends on two displacement terms w1,w2

w = (2 - 3/L.x)w1 + (-1 + 3/L.x)w2

∂w/∂t = ∂w/∂x.∂x/∂t = -3/L.w1.w1' + 3/L.w2.w2'

I'd imagine this wrong, any hints?

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# Differentiating a function of x wrt time

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