 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/d/solution

0 < x < π

lo g y = sin^{2} x = sin x > 0

. The chain rule gives

\frac{d z}{d x} = \frac{d z}{d y} \frac{d y}{d x} = \frac{1}{2 y s i n x} y sin 2 x .

Using

sin 2 x = 2 sin x cos x

yields

\frac{d z}{d x} = cos x .

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