Q.If $x^2+y^2=25$, find $\dfrac{dy}{dx}$.

Implicit Differentiation

When $y$ isn't alone

You can differentiate $y=x^2+3x$ term by term because $y$ is written explicitly in terms of $x$. But an equation like $x^2+y^2=25$, or $x^3+y^3=6xy$, does not give $y$ by itself — solving for $y$ is messy or downright impossible.

Implicit differentiation finds $\dfrac{dy}{dx}$ without isolating $y$: treat $y$ as an unknown function of $x$, differentiate the whole equation as it stands, then solve for $\dfrac{dy}{dx}$.

The one key move: $y$ is really $y(x)$

Wherever $y$ appears, picture $y(x)$ hiding inside. Differentiating a $y$-term therefore needs the chain rule, which tacks on a factor of $\dfrac{dy}{dx}$:

That extra $\dfrac{dy}{dx}$ on every $y$-term is the whole trick.

The procedure

Worked example

For $x^2+y^2=25$:

The answer naturally contains both $x$ and $y$ — that is normal here. To get the slope at a point on the curve, substitute the coordinates after differentiating; there is no need to solve for $y$ first. …