First part of the first one: use product rule: d(uv)/dx = u dv/dx + v du/dx.

Put u = x^2 and v = sin^4 x

How to do sin^4 x: put z = sin x and use the chain rule: dy/dx = (dy/dz) (dz/dx).

The rest of the examples are further applications of the chain and product rule. The book should explain it fully.

If you're wondering about the product rule, see where it's proved from first principles in the book (which I don't know but if it's any good it *must* have that in it).