What is the derivative of sin sin sin x

Sin [sin (sinx) ]

You have two ways into this the first is:

sin y

where y = sinz and z = sinx

d (siny) / dx = cosy * dy/dx

dy/dx = cosz * dz/dx

dz/dz = cosx

and do reverse sub

dy/dz = cosz * cosx

d (siny) / dx = cosz * cosx * cosy = cos (sin (sinx)) * cosx * cos (sinx)

or the way I prefer is take the derivative of each term in row:

cos (sin (sinx)) * cos (sinx) * cosx