When I write (Wx)y it means the partial derivative of W according to x with y constant !
Supposethat g(x,y)=c a constant and W=f(x,y,z) . Which of the following makes sense as the derivative Wx ? :
a) (Wx)x b) (Wx)y c) (Wx)z
2) suppose that cos(x-y)=5u and W=x^2*y*u. Find (Wu)x.
3) Consider the curve of points (x,y,z) satisfying
x^3-xyz=-1 and x^3+y^2*z^3-xz=7.
Use the method of differentials to find dz/dy at (x,y,z)=(1,1,2)
1. (c) (Wx)z makes sense.
But from g(x,y)=c, we can make derivative to x on both sides and get dg/dx+(dg/dy)*(dy/dx)=0, this implies dy/dx=-(dg/dx)/(dg/dy). So we plug it in (Wx)z=(df/dx)+(df/dy)*(dy/dx) and can find ...
This shows how to use the method of differentials.