implicit multivariable differentiation

[cheffy]

Find the value of dx/dz at the point (1, -1, -3) if the equation

\(\displaystyle \
xz + y\ln (x) - x^2 + 4 = 0
\\)

defines x as a function of the two independent variables y and z and the partial derivative exists.

I think I'm supposed to differentiate each part with respect to z, but I don't get how I would get the dx/dz. And should I move part(s) of that equation to the other side with the 0?

Thanks!

 

[arthur ohlsten]

xz+y ln[x] -x^2+4=0

find the rate of change of z with respect to x, with y as a parameter[or considered constant]

take derivative
d[xz]= x dz + z dx
d y[lnx]=[y/x] dx
d x^2=2xdx
d4=0

x dz +z dx+[y/x] dx -2x dx =0
divide by dz
x dz/dx +[z+[y/x]-2x]=0
dz/dx = [2x-z-[y/x]]/x
evaluate at 1,-1,-3
dz/dx=[2+3+1] / 1
dz/dx=6 answer

I believe this is what is wanted
Arthur