Find second derivative of

Question

f(x) = (x^5)(cosx)?

Answer 1

f (x ) = x ^5(Cosx )

Apply derivative on each side with respect of x .

Apply product rule in derivatives.

d/dx(uv ) = uv ' + vu '

u  = x ^5 , v = Cosx

f '(x ) = -x ^5 Sinx + Cosx 5x ^4

For second derivative ,again derivative on each sides with respect of x .

d/dx[f '(x )] = d/dx[- x ^5 Sinx  + Cosx 5x ^4]

f ''(x ) = - x ^5 Cosx - Sinx 5x ^4 +  Cosx 20x ^3 + 5x ^4(-Sinx)

f ''(x ) = - x ^5 Cosx - 5x ^4 Sinx + 20x ^3 Cosx - 5x ^4 Sinx

f ''(x ) = - x ^5 Cosx + 20x ^3 Cosx - 10x ^4 Sinx

f ''(x ) = - x ^3 (x ^2 Cosx - 20 Cosx + 10x Sinx )

f ''(x ) = - x ^3[Cosx (x ^2 - 20) + 10x Sinx ]

Second derivative of x ^5(Cosx) is -x ^3[Cosx(x ^2 - 20) + 10x Sinx ].