Question

Can someone please explain to me how these derivatives are calculated?

Answer 1

for reference:x = horizontal position of pendulum massy = vertical position of pendulum massθ = angle of pendulum (0 = vertical downwards, counter-clockwise is positive)L = length of rod (constant)

I understand everything up to the second set of equations.... can someone explain how the derivatives are taken?

Answer 2

Chain rule. If θ is a function of t, then dx/dt = dx/dθ * dθ/dt.

Answer 3

Because x and theta are unknown values, you have to assume that they aren't constants, thus x' and theta prime. The rest is just straight up conversions

Answer 4

not visualizing how the two pendulums are placed relative to one another but the derivatives follow from the usual rules given x1(theta1), y1(theta1), x2(theta2), y2(theta2), ie d/dx(sin(x))=cos(x) etc