 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/iv/solution

First express the function in the original variables:

f (x, y) = ln (1 + x^{2} + y^{2}) + ln (1 + x^{2} y^{2}) .

f_{x} = \frac{2 x}{1 + x ^{2} + y ^{2}} + \frac{2 x y ^{2}}{1 + x ^{2} y ^{2}} .

y

yields

f_{x y} = - \frac{4 x y}{( 1 + x ^{2} + y ^{2} ) ^{2}} + \frac{4 x y ( 1 + x ^{2} y ^{2} ) - 4 x ^{3} y ^{3}}{( 1 + x ^{2} y ^{2} ) ^{2}} .

The numerator in the second fraction reduces to

4 x y

, reproducing

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

Both logarithms are defined for every real

x, y

because their arguments are at least one.

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