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
!
.