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 ${\frac {f'(x)}{f(x)}}=\log(2+\cos x)+{\frac {x}{2+\cos x}}(-\sin x)$ so after rearranging, we obtain $\color {blue}f'(x)=(2+\cos(x))^{x}\left(\log \left(2+\cos(x)\right)-{\frac {x\sin(x)}{2+\cos(x)}}\right)$.