Let $S \subset \Bbb R^2$ given by $S = \{(x, y) : f(x)=xy^2\}$, where $f$ is a differentiable function on $\Bbb R$. Let $P=(a, b) \in S$. Write the equation of the tangent line to the curve at point $P$
Through implicit differentiation, I found that
$$f'(x) = y^2 + 2y' x y.$$
Inserting $(a, b)$, I found $f'(a) = b^2 + y'(a) ab$ leaving $y'(a)$ on one side, I got $$\frac{f'(a) - b^2}{a b}$$ For the equation of the tangent line, I used $(y-b) = m(x-a)$ where $m$ is the slope. However I got stuck trying to figure out whether or not I should sub in $f'(a)$ for $m$ or $y'(a)$ for $m$. Or is it something else completely? Any help is appreciated.