Question

I don't understand this proof using y = uv and
adding Δi. The proof using definition of derivative with limit I understood. Can someone help me?
(0) y = u v

(1) y + Δy = (u + Δu)(v + Δv) = uv + u Δv + v Δu + Δu Δv
--> (y + Δy) = f(x + Δx) - f(x) ?
--> (u + Δu) = g(x + Δx) - g(x) ?
--> (v + Δv) = h(x + Δx) - h(x) ?

(2) Δy = (y + Δy) - y = u Δv + v Δu + Δu Δv

(3) Δy/Δx = u Δv/Δx + v Δu/Δx + Δu Δv/Δx

(4) Δx -> 0

(5) δy/δx = u δv/δx + v δu/δx + 0 δv/δx
--> Why Δu goes to zero?
--> Why Δv did not goes to zero?

Answer 1

I understood that it is a intuitive graphical proof.

Answer 2

Afranio, the three equations you have under (1) of the form
(y + delta y) = f(x + delta x) - f(x) are incorrect.

This equation should be
Delta y= f(x + delta x) - f(x).

Similarly with the other two equations. So, as delta x goes to zero, f(x + delta x) approaches f(x). Therefore, delta y goes to zero as delta x goes to zero. Same thing with delta u and delta v. I hope this answers your first question.

For the second question, "Why doesn't delta v go to zero?", it does go to zero. However, it is being divided by delta x which also is going to zero. So, while delta v goes to zero, "delta v/ delta x" does not go to zero. It has a value.