Finding the Slope of a Curve Defined by an Implicit Function

Determining the slope of a curve defined by an implicit function can be a challenging task, but with the right approach, it can be done efficiently. In this article, we will explore the process of finding the slope of the curve defined by the equation x^3 - 3xy^2 + y^3 = 1 at the point (2, -1).

To begin, let's recall the formula for the slope of a curve defined by an implicit function F(x, y) = 0:

dy/dx = -∂F/∂x / ∂F/∂y

where ∂F/∂x and ∂F/∂y are the partial derivatives of F(x, y) with respect to x and y, respectively.

Now, let's apply this formula to the given equation:

Differentiate the equation with respect to x:

   ∂F/∂x = 3x^2 - 3y^2

Differentiate the equation with respect to y:

   ∂F/∂y = -6xy + 3y^2

Substitute the point (2, -1) into the partial derivatives:

   ∂F/∂x = 3(2)^2 - 3(-1)^2 = 12 - 3 = 9
   ∂F/∂y = -6(2)(-1) + 3(-1)^2 = 12 + 3 = 15

Plug the values into the slope formula:

   dy/dx = -∂F/∂x / ∂F/∂y
   dy/dx = -(9) / 15
   dy/dx = -3/5

Therefore, the slope of the curve defined by the equation x^3 - 3xy^2 + y^3 = 1 at the point (2, -1) is -3/5.

This process demonstrates the application of the implicit differentiation technique to find the slope of a curve defined by an implicit function. By carefully differentiating the equation and substituting the given point, we can determine the slope at that specific location on the curve.

Find the slope of the curve: x^3 - 3xy^2 + y^3 = 1 at the point (2, -1).

Finding the Slope of a Curve Defined by an Implicit Function