Calculating the Gradient:
To find the gradient of a function, we need to calculate the partial derivatives with respect to each variable. In this case, we have the function f(x, y) = x^2y^2 + xy + x.

Partial Derivative with respect to x:
To find the partial derivative of f with respect to x, we differentiate the function with respect to x while treating y as a constant. This gives us:
∂f/∂x = 2xy^2 + y + 1

Partial Derivative with respect to y:
Similarly, to find the partial derivative of f with respect to y, we differentiate the function with respect to y while treating x as a constant. This gives us:
∂f/∂y = 2x^2y + x

Gradient of the Function:
The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point. It is calculated by taking the partial derivatives of the function and combining them into a vector.

In this case, the gradient vector is:
∇f = (∂f/∂x, ∂f/∂y) = (2xy^2 + y + 1, 2x^2y + x)

Evaluating the Gradient at (1, 1):
To evaluate the gradient at a specific point, we substitute the values of x and y into the gradient vector. In this case, we are evaluating the gradient at (1, 1), so we substitute x = 1 and y = 1.

∇f(1, 1) = (2(1)(1)^2 + 1 + 1, 2(1)^2(1) + 1)
= (2 + 1 + 1, 2 + 1)
= (4, 3)

Magnitude of the Gradient:
The magnitude of a vector represents its length. In this case, we need to calculate the magnitude of the gradient vector at (1, 1).

The magnitude of a vector v = (a, b) is given by √(a^2 + b^2). Therefore, the magnitude of the gradient at (1, 1) is:
|∇f(1, 1)| = √(4^2 + 3^2)
= √(16 + 9)
= √25
= 5

Conclusion:
The magnitude of the gradient of the function f(x, y) = x^2y^2 + xy + x evaluated at (1, 1) is 5. This represents the rate of the steepest increase of the function at that point.