Calculate $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ at the points $(1,85,3)$ and $(1,85,-3)$, where $z$ is defined implicity by the equation $z^{4}+z^{2} x^{2}-y-5=0$
(Give exact answers. Use symbolic notation and fractions where needed.)

Step 1 ：We are given the equation \(z^{4}+z^{2} x^{2}-y-5=0\). We need to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the points \((1,85,3)\) and \((1,85,-3)\).

Step 2 ：We use the implicit function theorem to find these derivatives. The implicit function theorem states that if a relation implicitly defines one variable as a function of the others, then the derivatives of that implicit function can be found by differentiating the relation with respect to the other variables.

Step 3 ：Differentiating the given equation with respect to x, we get \(\frac{\partial z}{\partial x} = -\frac{2xz^{2}}{2x^{2}z + 4z^{3}}\).

Step 4 ：Differentiating the given equation with respect to y, we get \(\frac{\partial z}{\partial y} = \frac{1}{2x^{2}z + 4z^{3}}\).

Step 5 ：Substituting the point \((1,85,3)\) into these expressions, we find \(\frac{\partial z}{\partial x} = -\frac{3}{19}\) and \(\frac{\partial z}{\partial y} = \frac{1}{114}\).

Step 6 ：Substituting the point \((1,85,-3)\) into these expressions, we find \(\frac{\partial z}{\partial x} = \frac{3}{19}\) and \(\frac{\partial z}{\partial y} = -\frac{1}{114}\).

Step 7 ：So, the final answers are \(\boxed{-\frac{3}{19}, \frac{1}{114}, \frac{3}{19}, -\frac{1}{114}}\).