v=e^(xy)=v(x,y)
Given the concrete form of the function f (), the following results can be derived: namely, transform f(u, v) into the function g(x, y) of x, y:
z = f(u,v) = f [u(x,y),v(x,y)] = g(x,y)
Then calculate:
? z/? x =? g/? x
? z/? y =? g/? y
Since z is a function of (x, y), it is not dz/dx and dz/dy that need to be calculated, but the partial derivatives of z to x and y.
For example, suppose: f(u, v)=u+v //: that is, given a specific expression, it is easy to solve, and it is necessary to write a general expression.
More trouble!
u=x-y
v=e^(xy)
Then: z = f (u, v) = u+v = x-y+e (xy) = g (x, y)
So:? z/? x=? g/? x= 1+ y e^(xy)
? z/? y=? g/? y=- 1+x e^(xy)