Introduction:
In this problem, we are given a function f(x, y) = x^3 * y^3, where y represents a parabolic curve x^2 - 1. We need to find the total derivative of f with respect to x at x = 1.

Step 1: Finding the partial derivative of f with respect to x:
To find the total derivative of f with respect to x, we first need to find the partial derivative of f with respect to x. The partial derivative of f with respect to x can be found by differentiating f(x, y) with respect to x, while treating y as a constant.

Let's differentiate f(x, y) = x^3 * y^3 with respect to x:

∂f/∂x = ∂(x^3 * y^3)/∂x

Using the power rule of differentiation, we get:

∂f/∂x = 3x^2 * y^3

Step 2: Finding the partial derivative of f with respect to y:
Next, we need to find the partial derivative of f with respect to y. The partial derivative of f with respect to y can be found by differentiating f(x, y) with respect to y, while treating x as a constant.

Let's differentiate f(x, y) = x^3 * y^3 with respect to y:

∂f/∂y = ∂(x^3 * y^3)/∂y

Using the power rule of differentiation, we get:

∂f/∂y = 3x^3 * y^2

Step 3: Finding the total derivative of f with respect to x:
The total derivative of f with respect to x can be found using the chain rule. It is given by:

df/dx = ∂f/∂x + (∂f/∂y * dy/dx)

Since y represents a parabolic curve x^2 - 1, we can substitute y = x^2 - 1 in the partial derivatives ∂f/∂x and ∂f/∂y.

∂f/∂x = 3x^2 * (x^2 - 1)^3
∂f/∂y = 3x^3 * (x^2 - 1)^2

Next, we need to find dy/dx. Taking the derivative of y = x^2 - 1 with respect to x, we get:

dy/dx = 2x

Substituting the values of ∂f/∂x, ∂f/∂y, and dy/dx in the total derivative equation, we get:

df/dx = 3x^2 * (x^2 - 1)^3 + 3x^3 * (x^2 - 1)^2 * 2x

Simplifying this expression, we get:

df/dx = 3x^2 * (x^2 - 1)^3 + 6x^4 * (x^2 - 1)^2

Step 4: Evaluating the total derivative at x = 1:
To find the total derivative of f with respect to x at x =