Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 12 / a / ii

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- Solution ii

Solution

Differentiate the defining function with the other coordinates independent. Put

C = cosh (x + z)

for brevity. Then

F_{x} = yz - C, F_{y} = x z, F_{z} = x y - C .

The implicit differentiation ratios therefore give

(\frac{\partial x}{\partial y})_{z} = - \frac{x z}{yz - C}, (\frac{\partial x}{\partial z})_{y} = - \frac{x y - C}{yz - C}, (\frac{\partial y}{\partial z})_{x} = - \frac{x y - C}{x z} .

Each expression applies on the implicit surface wherever its own denominator is nonzero. The hyperbolic cosine appears because the derivative of

sinh (x + z)

with respect to either

x

or

z

is

cosh (x + z)

.

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