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Let x be a real number, and define f(x)=|log 2-sin x| and g(x)=f(f(x)). Then:
- g'(0)=-cos(log 2)
- g is differentiable at x=0 and g'(0)=-sin(log 2)
- g is not differentiable at x=0
- g'(0)=cos(log 2)
Correct answer: g'(0)=cos(log 2)
Solution
Near x=0, log2-sin x>0, so f(x)=log2-sin x and f'(x)=-cos x. Also log2-sin(f(x))>0 near 0, so g(x)=log2-sin(f(x)), giving g'(x)=-cos(f(x)) f'(x)=cos(f(x)) cos x. At x=0, f(0)=log2, so g'(0)=cos(log2).
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