By using Theorem 1, prove that Theorem 1Let f and g be differentiable vector-valued functi...

By using Theorem 1, prove that

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.

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2.

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3. for any constant c

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4.

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5.

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6.

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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.