Implicit Differentiation and product rule

[Loupster]

Hi,

So, I am reviewing Cal III, and I have come across something that I do not understand regarding implicit differentiation with partial derivatives:

x^3 + y^3 + z^3 + 6xyz = 1

implicit differentiation of z with respect to x:

3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0

*notive the (dz/dx) are partial derivatives, not regular derivatives

What I do not understand is that there are two '6xyz' derivatives. I understand how 6yz was formed, because it is wrt x, and the 6xy(dz/dx) because that is how the implicit part works, I believe. However, I do not understand how you can just add the variables twice, it seems like that would change this entire equality. . . ?

Any help would be great!
Thanks!

 

[lurflurf]

That comes from the product rule.
(6xyz)'=6(x)'yz+6xy(z')
where x'=1, but z' stays around