Question

During the lecture 4 of 18.01SC there a brief mention during the derivation of Quotient rule that u(x + Ꮩx) = u(x) + u(Ꮩx)

when u apply this to say product rule
u(x+Ꮩx) = u + Ꮩu
v(x+Ꮩx) = v + Ꮩv

gives u(x+Ꮩx)v(x+Ꮩx) = uv + uᏙv + vᏙu + ᏙuᏙv

taking limits we get with Ꮩx -> 0
lim Ꮩx-> 0 {Ꮩ(uv) / Ꮩx} =
u'v + v'u + lim Ꮩx ->0 ( (uv/Ꮩx) + (ᏙuᏙv / Ꮩx) )

here are the questions
(a) are for any two continuous, differentiable functions u& v
lim Ꮩx->0 (ᏙuᏙv / Ꮩx) = u'v(0) = u(0)v' ?

(b) since (uv)' = u'v+uv' so is
lim Ꮩx ->0 ( (uv/Ꮩx) + (ᏙuᏙv / Ꮩx) ) = 0 ?

Answer 1

What book do you use for this lecture?

Answer 2

I am just going with the lecture videos itself. And what ever I remember from my 12 grade maths