Find dy/dx if y = x^(cos x) + (cos x)^(sin x), 0 < x < π/2.

1. 1 · Write what is given and what is to be found. Given: y = x^(cos x) + (cos x)^(sin x). To find: dy/dx.
2. 2 · Split the sum. Let u = x^(cos x) and v = (cos x)^(sin x). Then y = u + v and dy/dx = du/dx + dv/dx.
3. 3 · Find du/dx.
   • log u = cos x log x
   • (1/u) du/dx = −sin x log x + (cos x)/x
   • du/dx = x^(cos x) [(cos x)/x − sin x log x]
4. 4 · Find dv/dx.
   • log v = sin x log(cos x)
   • (1/v) dv/dx = cos x log(cos x) + sin x × (−sin x/cos x)
   • dv/dx = (cos x)^(sin x) [cos x log(cos x) − sin x tan x]
5. 5 · Add the parts. dy/dx = x^(cos x) [(cos x)/x − sin x log x] + (cos x)^(sin x) [cos x log(cos x) − sin x tan x].
6. 6 · Check and box the answer. Each part has the form (original power) × [bracket]. Box the full expression.

Scoring tip: Never take the log of the whole sum. Split it into u and v first; this step alone carries a mark.