 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/iii/solution

For

f (u, v) = ln u + ln v

f_{v} = \frac{1}{v}, f_{uu} = - \frac{1}{u ^{2}}, f_{uv} = 0, f_{vv} = - \frac{1}{v ^{2}} .

Substituting these into part (ii) gives

f_{x y} = - \frac{4 x y}{u ^{2}} - \frac{4 x ^{3} y ^{3}}{v ^{2}} + \frac{4 x y}{v} .

Since

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

, the last two terms combine. Thus

f_{x y} = 4 x y [\frac{1}{( 1 + x ^{2} y ^{2} ) ^{2}} - \frac{1}{( 1 + x ^{2} + y ^{2} ) ^{2}}] .

 New to topics? Read the docs here!