Complicated implicit multivariable differentiation problem

[fogvajarash]

Homework Statement

Given that the surface $x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7$ has the equation z = f(x, y) in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find:

$\displaystyle\frac{\partial^{2} f}{\partial x^{2}}(1,1) = ?$

Homework Equations

The Attempt at a Solution

To make things easier, i have already found an expression for the partial derivative of z with respect to x:

$\displaystyle\frac{\partial f}{\partial x} = \displaystyle\frac{-6x^{5}y^{5}-7z^{9}x^{6}-4yz}{5y^{4}z^{4}+9z^{8}x^{7}+4xy}$

And at (1, 1), it's value is -17/18. I have tried to differentiate the expression with respect to x going from this general expression and doing so implicitly and then collecting the terms, however, i get two different results which are both wrong: 1129/729 and -160416. Is there an easier way to approach this problem or it is just tedious differentiation and being extremely careful with the terms?

 

[danago]

Should that first term on the denominator be 5y⁴z⁴ instead?

 

[fogvajarash]

Should that first term on the denominator be 5y⁴z⁴ instead?

Yes I'm sorry. I've fixed it but it won't affect the result anyways (as we have a y = 1 and we are differentiating with respect to x)