Proving Formula in Vector Differentiation

[ppoorrttee]

1. The problem statement
The formula that must be proven is:
d (A∙B) = A∙dB + dA∙B
du du du


2. The attempt at a solution
When I substitute the left side of the equation to the general formula of vector differentiation, I got the left side of the equation + ΔA∙ΔB
du
Now my only problem is that how can I prove that ΔA∙ΔB = 0 so that it will result to the above equation? Is there any identity in vector differentiation that is ΔA∙ΔB = 0?

 

[HallsofIvy]

First, what are $\Delta A$ and $\Delta B$? The "general formula of vector differentiation" that I know involves limits. What limits are you taking?