Yahoo Answers is shutting down on May 4th, 2021 (Eastern Time) and beginning April 20th, 2021 (Eastern Time) the Yahoo Answers website will be in read-only mode. There will be no changes to other Yahoo properties or services, or your Yahoo account. You can find more information about the Yahoo Answers shutdown and how to download your data on this help page.
Trending News
Please help with this question on continuous functions?
f(x,y) = ln(y-x)
u = e^(x+y) v=e^(x-y)
and f(x,y) = g(u,v) is a continous function show that using the chain rule
df/dy = u(dg/du) - v(dg/dv) and obtain a similar expression for df/dx
1 Answer
- Scarlet ManukaLv 710 years agoFavorite Answer
We don't actually need to know what f(x, y) is for this one. It's enough to know u and v in terms of x and y and to know that f(x, y) = g(u, v).
The chain rule says
∂f/∂y = ∂/∂y g(u, v) = ∂g/∂u ∂u/∂y + ∂g/∂v ∂v/∂y
= ∂g/∂u . e^(x+y) + ∂g/∂v . e^(x-y) . (-1)
= u ∂g/∂u - v ∂g/∂v.
Similarly
∂f/∂x = ∂/∂x g(u, v) = ∂g/∂u ∂u/∂x + ∂g/∂v ∂v/∂x
= ∂g/∂u . e^(x+y) + ∂g/∂v . e^(x-y)
= u ∂g/∂u + v ∂g/∂v.
