Higher-order derivatives? ?? in?d y?, ?y?,? ?and y? ?for the following functions.

Solution Step 1: Given y=sin x We have to determine y’,y’’,y’’’ For that we need following formulae d (sin x) = cos x dx d (x ) = nx n1 dx d (uv )=u dv + v du dx dx dx d u vdxu dx ( ) = 2 dx v v Step 2: y’= dy dx = d (sin x) dx d =cos x dx( ) 1 =cos x 2x y’ = cosx 2x d Now y’’= dx (y’) d cosx = ( ) dx 2x 2 x dx(co x)co xdx2x) = 2 (2x) 2 xdx(cosx)cosxdx2 ) = 2 (2x) 2 x (si x)d(x)cosx(21 ) = dx x 4x 2 x (si x2 x) co x(x) = 4x (sinx) (ox ) = 4x 1 cos = 4x [sin x + ( x )] =[ x sinx cosx] 4xx x sin x co x y’’ =[ 3 ] 4x2