Given the surface: z^2=x^2+xy^2z. Find dz/dx

To find dz/dx, we'll use partial differentiation. We'll differentiate the given surface equation with respect to x, while treating y as a constant.

Let's start by differentiating both sides of the equation:

d/dx(z^2) = d/dx(x^2+xy^2z)

Using the chain rule, we can differentiate the left side of the equation as follows:

2z * dz/dx = 2x + y^2z * dz/dx

Now, let's isolate dz/dx:

2z * dz/dx - y^2z * dz/dx = 2x

Factoring dz/dx:

dz/dx * (2z - y^2z) = 2x

Simplifying the expression:

dz/dx = 2x / (2z - y^2z)

So, dz/dx is equal to 2x divided by (2z - y^2z).