Find Tangent Line to Cardioid at (0, 0.5) Using Implicit Diff.

[winslow]

Homework Statement

Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5).

x² + y² = (2x² + 2y² - x)²

Homework Equations

Derivative rules
point slope formula

The Attempt at a Solution

I got

y' = [16x³-4x²+16xy²-4y²-4y²-8x²+2x] / [2y - 16x²y-16y³+8xy]Now the equation of the tangent line should come out to y = x + (1/2)

Not sure exactly how it gets that I know you use point slope formula once you find the slope but I'm not sure how to simplify that down

 

[Mark44]

Homework Statement

Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5).

x² + y² = (2x² + 2y² - x)²

Homework Equations

Derivative rules
point slope formula

The Attempt at a Solution

I got

y' = [16x³-4x²+16xy²-4y²-4y²-8x²+2x] / [2y - 16x²y-16y³+8xy]

I'll take your word that the above is correct. Now evaluate the right side at (0, 1/2). That gives the slope of the tangent line to the cardioid at that point.

After you have this value, use the point-slope form of the equation of a line to get the equation of the tangent line.

Now the equation of the tangent line should come out to y = x + (1/2)

Not sure exactly how it gets that I know you use point slope formula once you find the slope but I'm not sure how to simplify that down