On http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-c-parametric-equations-for-curves/session-21-keplers-second-law/MIT18_02SC_MNotes_k.pdf :
Middle of Page 1, under "1. Differentiation of Products of Vectors."

In the second way of proving the product rule, it says "Let t increase by /Delta t. Then r increases by /delta r, and s by/delta s, and the corresponding change in r (dot) s is given by
/delta r (dot) s =(r+\delta r) (dot) (s+\delta s) - r (dot) s.
..."

My question is where is the "\delta r (dot) s=..." equation from, how was it derived?

Question

On http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-c-parametric-equations-for-curves/session-21-keplers-second-law/MIT18_02SC_MNotes_k.pdf :
Middle of Page 1, under "1. Differentiation of Products of Vectors."

In the second way of proving the product rule, it says "Let t increase by /Delta t. Then r increases by /delta r, and s by/delta s, and the corresponding change in r (dot) s is given by
/delta r (dot) s =(r+\delta r) (dot) (s+\delta s) - r (dot) s.
..."

My question is where is the "\delta r (dot) s=..." equation from, how was it derived?

experimentx · Answer

this seems pretty obvious
Δ(r⋅s)=(r+Δr)⋅(s+Δs)−r⋅s = Δr.s + r.Δs + Δr.Δs
since Δr.Δs is very small ... i guess it get's ignored.