Derivatives of Implicit Functions

A function is explicit when $y$ is written directly in terms of $x$, as in $y = x^2 + 3$. It is implicit when $x$ and $y$ are tangled together in an equation that is not (or cannot easily be) solved for $y$, such as $x^2 + y^2 = 25$ or $x^2 + xy + y^2 = 7$.

Implicit differentiation finds $\dfrac{dy}{dx}$ without first solving for $y$: differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$, and then solve the resulting equation algebraically for $\dfrac{dy}{dx}$. The crucial point is that every $y$-term carries an extra $\dfrac{dy}{dx}$ factor by the chain rule:

Method — implicit differentiation:

1. Differentiate every term of the equation with respect to $x$.
2. For a pure $x$-term, differentiate normally; for any term containing $y$, apply the chain rule and attach $\dfrac{dy}{dx}$; for a mixed $xy$-term, use the product rule.
3. Collect all terms containing $\dfrac{dy}{dx}$ on one side and everything else on the other.
4. Factor out $\dfrac{dy}{dx}$ and divide to isolate it.

Illustration. For $x^2 + y^2 = 25$: differentiating gives $2x + 2y\dfrac{dy}{dx}=0$, so $\dfrac{dy}{dx}=-\dfrac{x}{y}$. The answer legitimately contains both $x$ and $y$ — that is normal for an implicit derivative.