Answer
\[-(\sec h\,x)\;(\tan h\,x)[2+\ln \sec h\,x]\]
Work Step by Step
It is given that \[y=\sec h \,x\;(1+\ln \sec h\, x)\;\;\;...(1)\]
Differentiate (1) with respect to $x$ using product rule
\[y'=(\sec h \,x)'\;(1+\ln \sec h\, x)+\sec h \,x\;(1+\ln \sec h\, x)'\]
We know that \[(\sec h\,x)'=-\sec h\,x\tan h\,x\]
\[\Rightarrow y'=(-\sec h\,x\tan h\,x)(1+\ln \sec h\, x)+\sec h\,x\left[\frac{1}{\sec h\,x}\cdot(-\sec h\,x \:\tan h\,x)\right]\]
\[y'=-\sec h\,x\tan h\,x-\sec h\,x\tan h\,x\ln \sec h\,x-\sec h\,x\tan h\,x\]
\[y'=-2\sec h\,x\tan h\,x-\sec h\,x\tan h\,x\ln\sec h\,x\]
\[y'=-\sec h\,x\tan h\,x[2+\ln \sec h\,x]\]
Hence , \[y'=-\sec h\,x\tan h\,x[2+\ln \sec h\,x]\]
