Chain Rule with Partial Derivative?

[mshiddensecret]

Homework Statement

Given that the surface x^7y^2+y^4z^6+z^8x^8+9xyz=12 has the equation z=f(xy) in a neighbourhod of the point (1,1,1) with f(x,y) differentiable, find the derivatives.

df/dx (1,1) = ?

d^2f/dx^2 (1,1) = ?

Homework Equations

The Attempt at a Solution

df/dx (1,1) I got -24/23 or
7x^6y^2+6y^4z^5(dz/dx)+8z^7x^8(dz/dx)+8z^8x^7+9yz+9xy(dz/dx)

d^2f/dx^2: I tried doing the same thing and squaring the derivative when they came up again but it didn't work.

 

[SteamKing]

When you take the partial derivative of f(x,y,z) w.r.t. x, you treat any y's or z's as constants.

For example, if f(x,y,z) = 3x²y + y²z, then

∂ f / ∂ x = 6xy

The chain rule would be used only if you know y and/or z as functions of x.