Implicit differentiation

Do the following for the curves below:

  1. Find \( \frac{dy}{dx} \).
  2. Check answers with WolframAlpha
  3. Find the tangent line to curve at the point given.
  4. Graph the curve and tangent line to check answer.

1. Eight curve: \( x^4 = 100(x^2 - y^2) \) at \( (x,y) = (5, \frac{5 \sqrt{3}}{2}) \).

Answers:


2. Eight curve (turned): \( y^4 = 100(y^2 - x^2) \) at \( (x,y) = (3, 3 \sqrt{10}) \).

Answers:


3. Devil's curve: \( y^2 (y^2 - 96) = x^2 (x^2 - 100) \) at \( (x,y) = (4, \sqrt{48+8 \sqrt{15}}) \).

Answers:


Challenge

C1. First Heart curve: \( (x^2+y^2-1)^3 = x^2 y^3 \). Find \( \frac{dy}{dx} \).

C2. Ampersand curve: \( (x-1) (2 x-3) (y^2-x^2) = 4 (x^2-2 x+y^2)^2 \). Find \( \frac{dy}{dx} \).

C3. Consider \( 4 x^3 - 2 y ^3 = x \). Find all the points where the tangent line is horizontal.

C4. Consider \( y^2 + xy + x^2 = 1 \). Find all the points where the tangent line is horizontal.

C5. Consider \( y^2 + xy + x^2 = 1 \). Find all the points where the tangent line is vertical.