Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 9

The product rule and chain rule first giveDifferentiating the first expression with respect to $y$ yields $e^{x^{2}y}(2y+7x^2{y}^2+2x^4{y}^3)$; differentiating the second with respect to $x$ gives exactly the same polynomial factor. ThusThis also follows from symmetry of second derivatives, since all derivatives of this function are continuous everywhere.

Use the fixed exponents $p,q$ in the original PDF. Assume positive $a,b$ so arbitrary real powers and a relative change are defined. Set $h=k/100$. With $b$ fixed, the exact new-to-old ratio is $(1+h{)}^p$. The binomial series, or the one-variable case of the second-order relative change of a monomial, givesTherefore the approximate percentage change through second order isFor negative $k$ this same signed expression describes a decrease.

Set $h=k/100$ and $j=m/100$. The exact ratio is $(1+h{)}^p(1+j{)}^q$. Multiplying the two binomial series and retaining total degree at most two gives the second-order relative change of a monomial:Multiply by one hundred to express the change as a percentage:The cross term accounts for the simultaneous changes in both inputs and cannot be recovered by simply adding their separate second-order percentage changes.