Question

Could someone explain the part (on http://web.mit.edu/wwmath/calculus/differentiation/chain-proof.html )

after it says "Differentiablility implies continuity; therefore..."
Why does
du→0
as
dx→0

Answer 1

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

Answer 2

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?