jacobian matrix. derivatives

holly123 · #1 $Old$ September 27th, 2009, 01:43 PM

calculate the jacobian matrix of the function at the given point. and write the formula for the total derivative.
M(x1,x2,x3,x4)= ( (x1+x2)/(x3+x4), (x3+x4)/(x1+x2) )
a= (1,1,1,1)

how do i set up the matrix for this? any hints would be greatly appreciated to start me off

holly123 · #2 $Old$ September 27th, 2009, 02:03 PM

okay so i set it up as a 2x4 matrix:
dM1/dx1 dM1/dx2 dM1/dx3 dM1/dx4
dM2/dx1 dM2/dx2 dM2/dx3 dM2/dx4

but from there i got confused. for example in the first one, how do i take the partial derivative of (x1+x2)/(x3+x4) with respect to x1? wouldn't it be like 1/0 ? then where does the point (1,1,1,1) come into play? ah i'm so confused..

Chris L T521 · #3 $Old$ September 27th, 2009, 02:28 PM

Quote:

Originally Posted by holly123 $View Post$

okay so i set it up as a 2x4 matrix:
dM1/dx1 dM1/dx2 dM1/dx3 dM1/dx4
dM2/dx1 dM2/dx2 dM2/dx3 dM2/dx4

but from there i got confused. for example in the first one, how do i take the partial derivative of (x1+x2)/(x3+x4) with respect to x1? wouldn't it be like 1/0 ? then where does the point (1,1,1,1) come into play? ah i'm so confused..

Think of $x_2,x_3,x_4$ as constants when differentiating with respect to $x_1$ . So $\frac{\partial}{\partial x_1}\left[\frac{x_1+x_2}{x_3+x_4}\right]\sim \frac{\,d}{\,dx_1}\left[\frac{x_1}{b}+\frac{a}{b}\right]=\frac{1}{b}$ . So $\frac{\partial}{\partial x_1}\frac{x_1+x_2}{x_3+x_4}=\frac{1}{x_3+x_4}$ .

Can you use this same idea to differentiate the others?

holly123 · #4 $Old$ September 27th, 2009, 04:10 PM

hmm okay so it would be

(1/x3+x4) (1/x3+x4) (x1+x2) (x1+x2)
(x3+x4) (x3+x4) (1/x1+x2) (1/x1+x2)

i think?

and you plug in (1,1,1,1) to get
1/2 1/2 2 2
2 2 1/2 1/2

is that right?