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
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
Scarlet Manuka
Favorite 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.