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