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

Question

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.

Answer 1

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.