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Question: can someone help plz let i be an interval let...

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Can someone help plz?

Let I be an interval, let c∈I and let f,g:I→ℝ be differentiable at c. By Caratheodory’s theorem there exist functions φ,ψ:I→ℝφ,ψ:I→R which are continuous at c and with φ(c)=f′(c)φ(c)=f′(c) and ψ(c)=g′(c)ψ(c)=g′(c)such that for all x∈I we have

f(x)=f(c)+φ(x)(x−c),g(x)=g(c)+ψ(x)(x−c).

Use these representations to prove the product rule i.e. prove that fg is differentiable at c and that

(fg)′(c)=f′(c)g(c)+f(c)g′(c).(fg)′(c)=f′(c)g(c)+f(c)g′(c).

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