Prove part 5 of Theorem 1.
Theorem 1
Let f and g be differentiable vector-valued functions of one variable, and let h be a differentiable scalar-valued function. Then the following identities hold.
1.
2.
3. for any constant c
4.
5.
6.
7.
Proof we prove part 4 as an illustration of the proof method for the rest.
Let f = f1i + f2j + f3k. Then
The steps are justified, respectively, by Theorem 1.10.2, the ordinary product rule, and definitions of vector arithmetic.
Proofs of the remaining parts are exercises.
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