Use implicit differentiation to find an equation of the tangent line to the curve, called a devil's curve, at the point (0, -2).

Question

Use implicit differentiation to find an equation of the tangent line to the curve, called a devil's curve, at the point (0, -2).

 

y² (y²-4) = x² (x² - 5)

 

 

Answer 1

Cool. Check it out:

 

https://www.desmos.com/calculator/g2cxnnuivf

 

To find y' it's simplest to remove the parentheses first:

 

y⁴ - 4y² = x⁴ - 5x²

 

Differentiate with respect to x.

 

4y³y' - 8yy' = 4x³ - 10x

 

y'(4y³ - 8y) = 4x³ - 10x

 

y' = (4x³ - 10x)/(4y³ - 8y)

 

This is in effect a formula for slopes of tangent lines to the graph of the original function.

 

To find the slope of the tangent line at (0,-2) plug x=0 and y = -2 into the "formula".

 

Since this results in 0/-16 = 0, the slope is 0 and therefore a horizontal line with a y value of -2, i.e. y = -2.