What is the third derivative of f(x)=xcos(x)

Question

f(x)=xcos(x)

Answer 1

The function f (x ) = x Cos(x )

Apply derivative on each side with respect of x .

d /dx (f (x )) = d /dx (x Cosx )

Apply product rule d /dx (uv ) = uv '+vu '.

f '(x ) = x (-Sinx ) + Cosx (1)

f '(x ) = - x Sinx + Cosx

Apply derivative on each side with respect of x .

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

f ''(x ) = d /dx ( - x Sinx ) + d /dx (Cosx )

f ''(x ) = - x Cosx + Sinx (-1) + (-Sinx )

f ''(x ) = - x Cosx - Sinx - Sinx

f ''(x ) = - x Cosx - 2Sinx

Again derivative on each side.

f '''(x ) = d /dx (- x Cosx - 2Sinx )

f '''(x ) = - x (-Sinx )+Cosx (-1) - 2Cosx

f '''(x ) = x Sinx - Cosx - 2Cosx

f '''(x ) = x Sinx - 3 Cosx