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I need to check differentiability for the function $\left|\sin x -1\right|$ where $x \in R$

Now clearly $\sin x \le 1, x \in R$ so the given function is equivalent to

$1 - \sin x$ and this function is everywhere differentiable on $\mathbb{R}$

But, according to my book function is non differentiable at $\dfrac{\pi}{2}$.

Can anyone please tell me what is wrong with my solution.

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Nothing is wrong with your solution. We have $|\sin x - 1| =1-\sin x$, and this is differentiable on all of $\Bbb R$. Assuming you've copied the problem correctly, and you've checked the answer key for the correct exercise, then your book is wrong. This happens from time to time.

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