 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/8/solution

Apply implicit differentiation, remembering that

y

depends on

x

1 + y^{'} + e^{x} + e^{y} y^{'} = 0.

Since

1 + e^{y} > 0

, division is always legitimate. Therefore

\frac{d y}{d x} = - \frac{1 + e ^{x}}{1 + e ^{y}} .

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