## Task Condition

Find the derivative.

$y=\lg \ln (\operatorname{ctg} x)$
## Solution

$y^{\prime}=(\lg \ln (\operatorname{ctg} x))^{\prime}=\frac{1}{\ln (\operatorname{ctg} x) \cdot \ln 10} \cdot(\ln (\operatorname{ctg} x))^{\prime}=$

$=\frac{1}{\ln (\operatorname{ctg} x) \cdot \ln 10} \cdot \frac{1}{\operatorname{ctg} x} \cdot \frac{-1}{\sin ^{2} x}=-\frac{1}{\ln (\operatorname{ctg} x) \cdot \ln 10 \cdot \operatorname{ctg} x \cdot \sin ^{2} x}=$

$=-\frac{2}{\ln (\operatorname{ctg} x) \cdot \ln 10 \cdot \sin 2 x}$

## Problem Kuznetsov Differentiation 8-18