 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/14/b/i/solution

The chain rule gives

f_{x} = f_{u} u_{x} + f_{v} v_{x}

and

f_{y} = f_{u} u_{y} + f_{v} v_{y}

. Since

u_{x} = 2 x

u_{y} = 2 y

v_{x} = 2 x y^{2}

and

v_{y} = 2 x^{2} y

, the answer is

f_{x} = 2 x f_{u} + 2 x y^{2} f_{v}, f_{y} = 2 y f_{u} + 2 x^{2} y f_{v} .

f

on the right is evaluated at

u = 1 + x^{2} + y^{2}

v = 1 + x^{2} y^{2}

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