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So the question is: Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

$$x^2 + y^2 = (5x^2 + 4y^2 − x)^2,\text{ at } (0, 1/4) \text{ (cardioid)}$$

If anyone can help me out I would like a step by step process to find the solution of this problem. I also don't really understand how to differentiate so if there is anything to help with that then it would be great.

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  • $\begingroup$ You just asked a similar question an hour ago and got an answer. Are you working your way through your homework exercises? $\endgroup$
    – John Wayland Bales
    Commented May 14, 2022 at 1:09
  • $\begingroup$ Yes. But just these two problems. And I realized that I don't really understand implicit differentiation at all and I'm hoping to get a more in depth explanation of how exactly to do that as well. The text book hasn't really helped that much either so I am here now. $\endgroup$
    – Mehmed Sahan
    Commented May 14, 2022 at 1:16

1 Answer 1

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Try working the problem by applying the following steps:

  1. Consider that both $x$ and $y$ are functions of a third variable $t$, then take $\dfrac{d}{dt}$ of both sides of the equation, being careful to apply power rule, product rule, etc.
  2. Multiply the result of step 1) by $dt$.
  3. Divide the result of step 2) by $dx$.
  4. Replace variables $x$ and $y$ by their values at the given point of tangency.
  5. Solve the resulting equation for $\dfrac{dy}{dx}$ to find the slope of the tangent line.
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  • $\begingroup$ Thank you. I screenshotted this to keep. $\endgroup$
    – Mehmed Sahan
    Commented May 14, 2022 at 1:30
  • $\begingroup$ I am assuming that once you have the slope of the tangent line you can find the equation of the tangent line. $\endgroup$
    – John Wayland Bales
    Commented May 14, 2022 at 1:36
  • $\begingroup$ I'm sure I can figure it out. $\endgroup$
    – Mehmed Sahan
    Commented May 14, 2022 at 1:37

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