Question

\[\text{Find} \frac{ \delta z }{ \delta x }\text{and} \frac{ \delta z }{ \delta y } \text{ if z is defined implicitly as a function of x and y by the equation}\]
\[x^3 + y^3 + z^3 + 6xyz = 1\]

Answer 1

To differentiate implicitly dz/dx with respect to x i would treat y as a constant right?

Answer 2

yes

Answer 3

then solve for dz/dx

Answer 4

Notation?

Answer 5

partial derivatives...

Answer 6

wait what I'm I doing here? I should be chillin... XD

Answer 7

\[3x^2 + 3z^2 \frac{ \delta z }{ \delta x } + 6yz + 6xy \frac{ \delta z }{ \delta x } = 0\]
\[\frac{ \delta z }{ \delta x } (3z^2 + 6xy) = 3(-x^2 -2 yz)\]
\[\frac{ \delta z }{ \delta x } = \frac{ 3(-x^2 - 2yz) }{ 3(z^2 + 2xy) }\]
\[ = \frac{ -x^2 - 2yz }{ z^2 + 2xy }\]

Answer 8

Are those partials?

Answer 9

yeah

Answer 10

what the? can't be that long