Let (x) = x sin x and g(x) = x cos x. (a) Show that (x) = g(x) + sin x and g'(x) = (x)

Let ƒ(x) = x sin x and g(x) = x cos x.
(a) Show that ƒ(x) = g(x) + sin x and g'(x) = −ƒ(x) + cos x.
(b) Verify that ƒ'(x) = −ƒ(x) + 2 cos x and g'(x) = −g(x) − 2 sin x.
(c) By further experimentation, try to find formulas for all higher derivatives of ƒ and g. The kth derivative depends on whether k = 4n, 4n + 1, 4n + 2, or 4n + 3.