Proving the formula for the directional derivatives of the of the sum and dot product

Define the directional derivative of a function at in the direction bywhenever the limit on the right exists.

Let and be functions with values in such that the directional derivatives and exist.

Prove that the sum and dot product have directional derivatives given byand

I've figured out how to prove the sum part already but I'm having trouble with the dot product part. I tried to do it backwards

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Re: Proving the formula for the directional derivatives of the of the sum and dot pro

Hint: Look at proving the product rule for normal, single variable derivatives. There's a term you add and subtract in the numerator to make everything cancel out.