Discussion
For \(x \in \mathbb{R}\),let \(f(x) = log3 - sinx\) and \(g(x) = f(f(x))\)Then g'(0)=
Answer: E
The value is \(g'(0) = cos(log3)\).
By the chain rule \(g'(x) = f'(f(x))\cdot f'(x)\), and \(f'(x) = -\cos x\). Also \(f(0) = \log 3 - 0 = \log 3\).
\[ \begin{aligned} g'(0) &= f'(\log 3)\cdot f'(0) \\ &= (-\cos(\log 3))(-\cos 0) = \cos(\log 3) \end{aligned} \]
\(-\cos(\log 3)\) forgets the second factor \(f'(0) = -1\), and the sine options differentiate \(\sin\) wrongly. \(2\cos(\log 3)\) adds the two factors instead of multiplying them.
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