Implicit Differentiation

Implicit differentiation finds $\frac{dy}{dx}$ when $y$ is defined implicitly by an equation in $x$ and $y$, rather than explicitly as $y = f(x)$.

The key rule: treat $y$ as a function of $x$, and apply the chain rule whenever you differentiate a term involving $y$:

$\frac{d}{dx}[y^n] = ny^{n-1} \cdot \frac{dy}{dx}$

Procedure:

1. Differentiate both sides with respect to $x$.
2. Apply the chain rule to every $y$ term (multiply by $\frac{dy}{dx}$).
3. Collect all $\frac{dy}{dx}$ terms on one side.
4. Solve for $\frac{dy}{dx}$.