let ( f(x)=2cosleft(\frac{x}{2}\right) ).\nfind ( f^{primeprime}(x) ).\nchoose 1 answer:\na (…

let ( f(x)=2cosleft(\frac{x}{2}\right) ).\nfind ( f^{primeprime}(x) ).\nchoose 1 answer:\na ( -\frac{1}{2}cosleft(\frac{x}{2}\right) )\nb ( -cosleft(\frac{x}{2}\right) )\nc ( -sinleft(\frac{x}{2}\right) )\nd ( -8cosleft(\frac{x}{2}\right) )

let ( f(x)=2cosleft(\frac{x}{2}\right) ).\nfind ( f^{primeprime}(x) ).\nchoose 1 answer:\na ( -\frac{1}{2}cosleft(\frac{x}{2}\right) )\nb ( -cosleft(\frac{x}{2}\right) )\nc ( -sinleft(\frac{x}{2}\right) )\nd ( -8cosleft(\frac{x}{2}\right) )

Answer

Explanation:

Step1: Find the first derivative

Use the chain rule. The derivative of (\cos(u)) with (u = \frac{x}{2}) is (-\sin(u)\cdot u'). Here (u'=\frac{1}{2}). (f'(x)=2\times(-\sin(\frac{x}{2}))\times\frac{1}{2}=-\sin(\frac{x}{2}))

Step2: Find the second derivative

Again use the chain rule. For (y = -\sin(u)) with (u=\frac{x}{2}), (y'=-\cos(u)\cdot u') and (u'=\frac{1}{2}). (f''(x)=-\cos(\frac{x}{2})\times\frac{1}{2}=-\frac{1}{2}\cos(\frac{x}{2}))

Answer:

A. (-\frac{1}{2}\cos\left(\frac{x}{2}\right))