Question

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. The curve is represented by the equation y²(y² - 4) = x²(x² - 5) and the point is (0, -2) (devil's curve).

Answer 1

The equation of the tangent line to Devil's Curve at the point (0, -2) can be found using implicit differentiation to obtain the slope and then applying it to the point-slope form equation.

Step-by-step explanation:

To find the equation of the tangent line to Devil's Curve at the point (0, -2) using implicit differentiation, we first differentiate both sides of the given equation with respect to x. This process will give us the derivative dy/dx, which represents the slope of the tangent line at any point on the curve.

We then evaluate this derivative at the given point to find the specific slope. Once we have the slope, we can use the point-slope form equation to find the equation of the tangent line:

y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope found from differentiation.

Since the point is (0, -2), our equation becomes:

y + 2 = m(x - 0) or y = mx - 2, where m is the specific slope of the tangent at (0, -2).