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This is from Lecture Session 10 of MIT OCW Math 18.01 Scholar - Single Variable Calculus.

How does the Delta x term get into the numerator of the numerator?

\[{{1\overΔx}(Δu)v−u(Δv)\over(v+Δv)v} \to {{Δu \overΔx}v−u{Δv \overΔx}\over(v+Δv)v}\]is the step?

And then ended up as you have on the right hand side.

see attached for more clarity. It's near the bottom third of the page...

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