Partial derivatives for implicit functions

[Srikanth]

Hi
I have a function f(x,y,z)=0. I would like to find dx/dy and dx/dz
(where I am using d for partial derivatives, I'm not sure how to get
the actual symbol in this edit box), where both would be functions of
x, y and z. Unfortunately, I cannot get an explicit solution x=g(y,z).
Any ideas on how I'd go about solving this?
Thanks
Srikanth

[dh]

Hi,

according to the impicite function theorem, we get:

dx/dy= -fy/fx and dx/dz= -fz/fx

Therefore, using D we get:

-D[f[x, y, z], y]/D[f[x, y, z], x]

and

-D[f[x, y, z], z]/D[f[x, y, z], x]

, Daniel

[Jens-Peer Kuska]

Hi,

df = Dt[f[x, y, z]] == 0 /. {Dt[x] -> dx, Dt[y] -> dy, Dt[z] -> dz} ;

(x'[y] == (dx /dy /. Solve[df , dx] // Expand )) /. dz :> z'[y]*dy

?

Regards
Jens

[David Park]

Srikanth,

Use implicit differentiation. Copy and evaluate the following in one cell.

Print["Find dy/dx for the equation:"]
eqn1 = 2 y == x^2 + Sin[y]
Print["To calculate dy/dx implicitly write equation with y as a \
function of x."]
eqn2 = eqn1 /. y -> y[x]
Print["Differentiate with respect to x"]
step1 = D[eqn, x]
Print["Solve the equation for y'[x]."]
step2 = Equal @@ Part[Solve[step1, y'[x]], 1, 1]
Print["Change y[x] back to y."]
step3 = step2 /. y[x] -> y
Print["Get rid of the extra minus signs."]
MapAt[Minus, step3, {{2, 1}, {2, 3, 1}}]

If you have Presentations you could also get rid of the minus signs with:

Needs["Presentations`Master`"]

MapAt[MultiplyByOne[-1], step3, 2]

which multiplies numerator and denominator by -1.

David Park
djm...@comcast.net
http://home.comcast.net/~djmpark/

[Srikanth]

Thanks a lot - implicit differentiation gave me the result I needed.