Question

I'm having trouble understanding the Quotient Rule from Session 10.

If delta_u = u(x + delta_x) - u(x), and
if delta_v = v(x + delta_x) - v(x),

how does the numerator simplify to

(u + delta_u / v + delta_v) + u(x)/v(x)

in the first step?

And, how is

u(x + delta_x) = u(x) + delta_u

used?

Answer 1

they defined delta_u as
delta_u = u(x + delta_x) - u(x)

once you have introduce delta_u, you can solve for u(x + delta_x)
i.e. u(x + delta_x) = u(x)+delta_u
they then use this result to replace u(x + delta_x) with u(x)+delta_u
similarly for v.

For what it's worth, here is their derivation