 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/18/f/solution

Substitution first makes the square-root branch transparent:

z = lo g (e^{s i n^{2} x}) = sin^{2} x = ∣ sin x ∣.

On the given interval this is exactly

sin x

, so direct differentiation gives

z^{''} = - sin x,

in agreement with the chain-rule calculation. Outside this interval one must retain the absolute value; there are cusps at integer multiples of

π

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