Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / F / F2 / b / i

Table of contents
- Solution i

Solution

Put

Q = 1 + x^{2} + y^{2} > 0

. Differentiation of

Q^{- 1}

gives

\nabla f (x, y) = \frac{- 2}{( 1 + x ^{2} + y ^{2} ) ^{2}} (x y) .

The positive denominator also shows that the function and these derivatives are smooth throughout

R^{2}

.

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