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Question:

For x ∈ R, f(x) = |log₂ - sinx| and g(x) = f(f(x)), then:

g is differentiable at x=0 and g'(0) = -sin(log₂)

g'(0) = cos(log₂)

g'(0) = -cos(log₂)

g is not differentiable at x=0

Solution:

g(x) = f(f(x))
g'(x) = f'(f(x))f'(x)
g'(0) = f'(f(0))f'(0)
As x tends to 0, log₂ > sinx
f(x) = log₂ - sinx
f'(x) = -cosx
f'(0) = -1
f'(log₂) = -cos(log₂)
g'(0) = (-cos(log₂))(-1) = cos(log₂)