# How would I use implicit differentiation to find an equation of the tangent line to the curve at the given point.

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.

• You just asked a similar question an hour ago and got an answer. Are you working your way through your homework exercises? Commented May 14, 2022 at 1:09
• 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. Commented May 14, 2022 at 1:16

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.