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#1 2007-07-29 16:49:54

meebo0129
Member
Registered: 2007-07-14
Posts: 16

Partial derivatives

If anyone could explain step by step how following problem is done, it would be greatly appreciated!

If F(x,y,z)=0 defines implicitly z = f(x,y), then ∂z/∂x = -F_x/F_z and ∂z/∂y = -F_y/F_z.

Find ∂z/∂x and ∂z/∂y as functions of x, y, and z, assuming that z=f(x,y) satisfies the given equation:

x^3 + y^3 + z^3 = xyz

Last edited by meebo0129 (2007-07-30 00:38:18)

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#2 2007-08-01 15:51:46

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Partial derivatives

just set
F(x,y,z)=x^3 + y^3 + z^3 -xyz


X'(y-Xβ)=0

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#3 2007-08-02 20:31:14

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Partial derivatives

meebo0129 wrote:

If F(x,y,z)=0 defines implicitly z = f(x,y), then ∂z/∂x = -F_x/F_z and ∂z/∂y = -F_y/F_z.

Differentiate F implicitly wrt x keeping y constant, using the chain rule for partial derivatives.

Now differentiate F implicitly wrt y keeping x constant in the same way.

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