The slope of the curve y³ - xy² = 4 at the point where y = 2 is

Solution:

Given y³ - xy² = 4 and the point y = 2

Slope of the curve is given by the derivative of the given function

First, we need to solve for x, put y = 2 in the given equation

⇒ 2³ - x(2²) = 4

⇒ 8 - 4x = 4

⇒ 4x = 8 - 4

= 4x = 4

⇒ x = 1

Point is (1, 2)

Now, we shall perform the implicit differentiation

In implicit differentiation , we find the derivative of one variable keeping the other constant

3y²dy/dx - y² - 2xy dy/dx

dy/dx{3y² - 2xy} = y²

dy/dx = y²/ [3y² - 2xy]

Slope at (1, 2)

dy/dx = 4/[12-4]

= 4/8

= 1/2

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The slope of the curve y³ - xy² = 4 at the point where y = 2 is

Summary:

The slope of the curve y³ - xy² = 4 at the point where y = 2 is 1/2.