Question

Let F(u, v) be a function of two variables. Suppose that u = x + y and v = x y. Express ∂ 2 F /∂ x ∂ y in terms of u- and v-derivatives of F.

Answer 1

We first compute ∂ F /∂ x and ∂ F /∂ y

∂ F /∂ x = ∂ F /∂ u + y ∂ F /∂ v

∂ F /∂ x = ∂ F /∂ u + x ∂ F/ ∂ v

Note that these formulas are valid for all functions F(u, v). In particular we can replace F by ∂ F/ ∂ y

When calculating ∂ /∂ x (x81 x ∂ F /∂ v ) we had to apply the product rule. Differentiating the first factor gives ∂ /∂ x (x) = 1 and for the second factor we use the result from above, which was obtained using the chain rule