Implicit differentiation: when y cannot be freed

Most curves you've met are explicit: $y$ is on its own, like $y = x^2 + 3$.

An implicit equation has $x$ and $y$ mixed up, like

$x^2 + y^2 = xy + 7$

You could try to rearrange it to get $y$ on its own, but that's often messy or impossible. Implicit differentiation lets you find the gradient without rearranging.

$y$ is still a function of $x$: as $x$ changes, $y$ changes. So a term like $y^2$ is really "(something that depends on $x$) squared".

That means the chain rule applies. Differentiate it as if $y$ were the variable (so $y^2$ gives $2y$), then multiply by $\dfrac{dy}{dx}$:

$\frac{d}{dx}(y^2) = 2y\,\frac{dy}{dx}$

More examples:

• $\dfrac{d}{dx}(y) = \dfrac{dy}{dx}$
• $\dfrac{d}{dx}(y^3) = 3y^2\,\dfrac{dy}{dx}$
• $\dfrac{d}{dx}(\sin y) = \cos y\,\dfrac{dy}{dx}$

Terms with only $x$ in them are differentiated exactly as before. No extra factor.

A term like $xy$ is a product, so use the product rule. With $u = x$ and $v = y$:

$\frac{d}{dx}(xy) = x\,\frac{dy}{dx} + y \times 1 = x\,\frac{dy}{dx} + y$

The $\dfrac{dy}{dx}$ appears because we differentiated the $y$.

1. Differentiate every term on both sides with respect to $x$.
2. Attach $\dfrac{dy}{dx}$ every time you differentiate something with $y$ in it.
3. Collect all the $\dfrac{dy}{dx}$ terms on one side, and everything else on the other.
4. Factorise out $\dfrac{dy}{dx}$.
5. Divide to get $\dfrac{dy}{dx}$ on its own.

Constants differentiate to 0, as usual.