Science:Math Exam Resources/Courses/MATH100/December 2018/Question 01 (e)/Solution 1

Taking the logaritm of the orignal function, we have $\log f=x\log (2+\cos x)$, Computing its derivative (using the product rule for the right hand side) gives $\frac{f^{\prime }(x)}{f(x)}=\log (2+\cos x)+\frac{x}{2+\cos x}(-\sin x)$ so after rearranging, we obtain $f^{\prime }(x)=(2+\cos (x){)}^x(\log (2+\cos (x))-\frac{x\sin (x)}{2+\cos (x)})$.