Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 6A / b / i

Table of contents
- Solution i

Solution

The partial derivatives of

f

are

f_{x} = 2 x

,

f_{y} = 4 y (a - y^{2})

and

f_{z} = 2 z

. Therefore the three implicit differentiation formulas are

(\frac{\partial x}{\partial y})_{z} = \frac{2 y ( y ^{2} - a )}{x}, (\frac{\partial y}{\partial z})_{x} = \frac{z}{2 y ( y ^{2} - a )}, (\frac{\partial z}{\partial x})_{y} = - \frac{x}{z} .

Their product is

- 1

wherever

x yz (y^{2} - a) \neq = 0

. The excluded factors identify exactly where these quotient formulas cannot all represent finite partial derivatives.

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