Implicit partial derivative

Is this the complete problem statement? Generally speaking you don't find "∂z/∂x" of something, and more than you find "dy/dx" of something. You can, however, find the derivative with respect to, say, x of a function (denoted d/dx(f)) or the partial of some function with respect to, say z (denoted ∂/∂x(f).

The symbols ∂z/∂x and dy/dx represent derivatives, whereas the symbols ∂/∂x and d/dx represent operators that can be applied to functions.

Please verify that what you're providing is the complete problem statement.

 

Corrected to: find (∂z/∂x) of if x³ + y³ + z³ + 6xyz = 1

It looks to me like there's a mistake in the solution guide.
##∂/∂x(x^3) = 3x^2## and ##∂/∂x(y^3) = 0## and ##∂/∂x(z^3) = 3z^2 ∂z/∂x##

BUT
##∂/∂x(6xyz) = 6yz## if z is independent of x, in which case ∂z/∂x would be 0. However, if z is dependent on x, you need to use the product rule. Since you're asked to find ∂z/∂x, it must be the case that z is a function of x (is dependent on x).