⋅ here means dot product
Here is a less coordinate-dependent "proof" of the same formula, (x(t)⋅y(t))=x(t)⋅y'(t)+x'(t)⋅y(t)... (1):

For small values of h, we have x(t + h) ≈ x(t) + hx'(t);
and similarly y(t + h) ≈ y(t) + hy'(t):
So for small h,
x(t + h)⋅y(t + h) ≈(x(t) + hx'(t))⋅(y(t) + hy'(t))
≈ x(t)⋅y(t) + h(x(t)⋅y'(t) + x'(t)⋅y(t));

and so
(x(t + h)⋅y(t + h)- x(t)⋅y(t))/h ≈ x(t)⋅y'(t) + x'(t)⋅y(t).

Taking the limit as h --> 0, the approximately equals sign "≈" changes into "=", and we get (1), which is the expression at the beginning.

So, I need to make this "proof" into a proof. In particular, your argument should use limits in a rigorous way, and not require making use of the ≈ symbol, and please give me help on how to do this problem because I really have no idea...

Obviously, you did not use 'preview' to see that your post did not display correctly.

If this was copy and pasted, then that is the reason.

Unfortunately, there is no preview feature in the GO skin :(

western50, can you re-type your question manually, and if you can learn use the LaTeX and use them in your posts, that would be most helpful in such cases.

There is no typo in the whole problem, I did not just copy and paste, I typed it up.

It is not displaying correctly on my end. I see a bunch of squiggly nonsensical symbols. LaTex would be the ticket. For one, your 'approximately' sign is not showing.