Second derivative of?

Question

( x ^ (1/2) ) * (cosx)

Answer 1

Given that √x cos x

First derivative of √x cos x =  d ⁄ dx ( √x cos x )

                                               = (d ⁄ dx √x) cos x + √x (d ⁄ dx cos x )

                                               = 1 ⁄ (2√x) cos x + √x (-sin x)

                                               = 1 ⁄ (2√x) cos x - √x sin x

Second derivative = d ⁄ dx ( 1 ⁄ (2√x) cos x - √x sin x )

                                 = d ⁄ dx ( 1 ⁄ (2√x) cos x ) - d ⁄ dx ( √x sin x )

                                 = [(d ⁄ dx 1 ⁄ (2√x)) cos x + 1 ⁄ (2√x) d ⁄ dx ( cos x ) ] - [  (d ⁄ dx √x) sin x + √x (d ⁄ dx sin x) ]

                                 = [ -1 ⁄ (4√[x³]) cos x + 1 ⁄ (2√x) (-sin x) ] - [ 1 ⁄ (2√x) sin x + √x (cos x) ]

                                 = [ -1 ⁄ (4√[x³]) cos x - 1 ⁄ (2√x) sin x ] - [ 1 ⁄ (2√x) sin x + √x cos x ]

                                 = -1 ⁄ (4√[x³]) cos x - 1 ⁄ (2√x) sin x - 1 ⁄ (2√x) sin x - √x cos x

                                 = -1 ⁄ (4√[x³]) cos x - 2/(2√x) sin x - √x cos x

                                 = (-1⁄ (4√[x³])- √x) cosx -1/√x sin x

                                 = -(1⁄ (4√[x³]) + √x)cosx -1/√x sin x

                                 = -(1+ 4√x√x³)/(4√[x³])cosx -1/√x sin x

                                 =  - [(1+ 4x²)  ⁄  (4x√x)] cosx -1/√x sin x

                                 = -[1 ⁄ √x] sin x - [(4x²+1) ⁄ (4x√x)] cos x

Therefore the second derivative of √x cos x is -[1 ⁄ √x] sin x - [(4x²+1) ⁄ (4x√x)] cos x.