Multivariable Calculus: Partial Derivatives

Consider the surface F(x,y,z) = x^7z^7 + sin(y^5z^7 ) + 1 = 0.
Find the following partial derivatives: dz/dx and dz/dy.

I don't know how to take the derivative of z w.r.to. x or y because that variable is paramater for the surface F. I know how to do dF/dx or dF/dz. Please explain.

Notice that the total derivative of F with respect to x is (this total derivative d/dx as opposed to partial derivative ∂/∂x)

dF/dx = (∂F/∂x) dx/dx + (∂F/∂y) ∂y/∂x + (∂F/∂z) ∂z/∂x.

Since F is constant--you have a level surface, and as y is independent of x, you get

0 = (∂F/∂x) + (∂F/∂z) ∂z/∂x ==> ∂z/∂x = - (∂F/∂x)/(∂F/∂z).

The same games shows that

∂z/∂y = - (∂F/∂y)/(∂F/∂z).

So, for example,

∂z/∂x = - (7x^6z^7)/(7x^7z^6 + 7y^5z^6cos(y^5z^7)).

It should be straight forward for you to compute ∂z/∂y.

x^7 z^7 + sin(y^5 z^7 ) + 1 = 0

Implicit differentiation with respect to x (y constant)
7 x^6 z^7 + 7 x^7 z^6 ∂z/∂x + cos(y^5 z^7)*7 y^5 z^6 ∂z/∂x = 0
x^6 z^7 + x^7 z^6 ∂z/∂x + cos(y^5 z^7)* y^5 z^6 ∂z/∂x = 0
x^7 z^6 ∂z/∂x + cos(y^5 z^7)* y^5 z^6 ∂z/∂x = -x^6 z^7
[x^7 +y^5 cos(y^5 z^7)] z^6 ∂z/∂x = -x^6 z^7
∂z/∂x = -z x^6 /[x^7 + y^5 cos(y^5 z^7)]

Implicit differentiation with respect to y (x constant)
7 x^7 z^6 ∂z/∂y + cos(y^5 z^7)*(5 y^4 z^7 + 7 y^5 z^6 ∂z/∂y) = 0
7 x^7 z^6 ∂z/∂y + cos(y^5 z^7)*(7 y^5 z^6 ∂z/∂y) = -5 y^4 z^7 cos(y^5 z^7)
7 z^6 [x^7 + y^5 cos(y^5 z^7)] ∂z/∂y = - 5 y^4 z^7 cos(y^5 z^7)
∂z/∂y = -5 z y^4 cos(y^5 z^7)/7 [x^7 + y^5 cos(y^5 z^7)]