 Solution

ID: past-exam-of-the-options-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/f2/a/iii/solution

F : U \subseteq R^{n} \to R^{m}

is differentiable at

x

, and

G

is defined on a neighborhood of

F (x)

and differentiable there, the chain rule states

D (G \circ F)_{x} = D G_{F (x)} \circ D F_{x} .

In standard coordinates the same statement is

J_{G \circ F} (x) = J_{G} (F (x)) J_{F} (x) .

The evaluation point in the first factor is

F (x)

, and the matrix order follows the order in which the linear maps act on increments.

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