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18.022 Recitation Quiz (with solutions) 20 October 2014 1. Suppose that f : R2 → R2 and g : R2 → R2 are differentiable. Find the total derivative D ( f ◦ g). (Note: you may write f as ( f 1 , f 2 ), where f i : R2 → R for i ∈ {1, 2}, and similarly for g.) Solution. Let’s use the variables s and t for the arguments of g, and let’s use the variables x and y for the arguments of f . The composition f ◦ g is given by ( f ◦ g)(s, t) = ( f 1 ( g1 (s, t), g2 (s, t)), f 2 ( g1 (s, t), g2 (s, t))). By the chain rule, the partial derivative of f 1 ( g1 (s, t), g2 (s, t)) with respect to s is ∂ f 1 ∂g1 ∂ f ∂g2 + 1 . ∂x ∂s ∂y ∂s Doing similar calculations for the other terms, we end up with ( D ( f ◦ g))(s, t) = ∂ f1 ∂x ∂ f2 ∂x ∂g1 ∂s ∂g1 ∂s + + ∂ f1 ∂y ∂ f2 ∂y ∂g2 ∂s ∂g2 ∂s ∂ f1 ∂x ∂ f2 ∂x ∂g1 ∂t ∂g1 ∂t + + ∂ f1 ∂y ∂ f2 ∂y ∂g2 ∂t ∂g2 ∂t ! .