 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2003/calculus-probability-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/2/a/i/solution

Assume all partial derivatives needed below are continuous. The multivariable chain rule is

f_{x} = g_{u} u_{x} + g_{v} v_{x},

g

are evaluated at

(u (x, y), v (x, y))

f_{xx} = (g_{uu} u_{x} + g_{uv} v_{x}) u_{x} + g_{u} u_{xx} + (g_{vu} u_{x} + g_{vv} v_{x}) v_{x} + g_{v} v_{xx} .

g_{vu} = g_{uv}

, so

f_{xx} = g_{u} u_{xx} + g_{v} v_{xx} + g_{uu} u_{x}^{2} + 2 g_{uv} u_{x} v_{x} + g_{vv} v_{x}^{2} .

Differentiating

f_{x}

with respect to

y

instead gives

f_{x y} = g_{u} u_{x y} + g_{v} v_{x y} + g_{uu} u_{x} u_{y} + g_{uv} (u_{x} v_{y} + v_{x} u_{y}) + g_{vv} v_{x} v_{y} .

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