Could someone explain the part (on http://web.mit.edu/wwmath/calculus/differentiation/chain-proof.html )
after it says "Differentiablility implies continuity; therefore..."
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hard to follow the proof on that but:
think of 2 gears that mesh together at the edges; as one rotates the other rotates.
We can call the main gear U, and the gear that U spins is then Y
The energy to spin this setup is X.
As the change in X goes to zero, the speed of gear U goes to 0; and since gear Y depends on U; as U goes to 0 so does Y. This is the "chain" rule.
Y depends on U: y(u(x))
U depends on X: u(x)
the change in Y depends in X; dy/dx thru U:
dy/dx = dy/du du/dx
if Y is not contiuous with U, then there is a point where Y is not depending on U for its change in momentum right?