The tangent plane of the surface 5𝑥𝑥𝑦𝑦 + 𝑦𝑦 2 + 1 = 𝑧𝑧 at the point (1,1,7) is

To find the equation of the tangent plane to the surface at a given point, we first need to find the gradient of the function. The gradient is a vector that points in the direction of the greatest rate of increase of the function, and its components are the partial derivatives of the function with respect to each variable.

The given surface is defined by the function f(x, y, z) = 5xy + y^2 + 1 - z = 0.

We first compute the partial derivatives of f with respect to x, y, and z:

∂f/∂x = 5y, ∂f/∂y = 5x + 2y, ∂f/∂z = -1.

We evaluate these at the point (1,1,7):

∂f/∂x(1,1,7) = 5, ∂f/∂y(1,1,7) = 7, ∂f/∂z(1,1,7) = -1.

The gradient of f at (1,1,7) is then the vector (5, 7, -1).

The equation of the tangent plane at (1,1,7) is given by

5(x - 1) + 7(y - 1) - 1(z - 7) = 0.

Simplifying this gives the equation of the tangent plane:

5x + 7y - z = 5.