What is the differentiation of log sin x ?

Solution :

We have, y = log sin x

Let u = sin x \(\implies\) \(du\over dx\) = cos x

And, y = log u \(\implies\) \(dy\over du\) = \(1\over u\)

Now, \(dy\over dx\) = \(dy\over du\) \(\times\) \(du\over dx\)

\(\implies\) \(dy\over dx\) = \(1\over u\) \(\times\) cos x

\(\implies\) \(dy\over dx\) = \(1\over sin x\) \(\times\) cos x = cot x

Hence, the differentiation of log sin x with respect to x is cot x.