If `y=cos^(2)x^(2)`, find `(dy)/(dx)`.

Explanation

`y = (cos x^(2))^(2)`. Put `x^(2) = t` and `cosx^(2) = cos t = u`, so that
`y = u^(2), u = cost ` and `t = x^(2)`
`:. (dy)/(du) = 2u, (du)/(dt) = - sint` and `(dt)/(dx) = 2x`.
So, `(dy)/(dx) = ((dy)/(du) xx (du)/(dt) xx (dt)/(dx))`
`= - 4ux sin t = - 4x sint cos t [ :' u = cos t]`
`= - 4x sin x^(2) cos x^(2) = -2x sin (2x^(2)) [:' t = x^(2)]`.