AHL 5.14—Implicit functions, related rates, optimisation

Most differentiation so far has involved explicit functions — equations where $y$ is written directly as a function of $x$, like $y=x^2+3x$. But many important curves and relationships cannot be (or are not conveniently) written this way.

Examples of implicitly defined curves:

• Circle: $x^2+y^2=25$
• Ellipse: $\frac{x^2}{9}+\frac{y^2}{4}=1$
• Folium of Descartes: $x^3+y^3=3xy$

In these cases, isolating $y$ may be messy or impossible. Implicit differentiation lets us find $\frac{dy}{dx}$ directly, without rearranging.

The key insight: we still differentiate both sides with respect to $x$, but whenever we differentiate a term involving $y$, we apply the chain rule — because $y$ is assumed to be a function of $x$.

The entire machinery of implicit differentiation rests on the chain rule. Recall:

$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))\cdot g^{\prime }(x)$

When we treat $y$ as an unknown function of $x$, differentiating $y^n$ with respect to $x$ gives:

$\frac{d}{dx}(y^n)=n{y}^{n-1}\cdot \frac{dy}{dx}$

This extra $\frac{dy}{dx}$ factor is what students most often forget. By contrast, differentiating $x^n$ simply gives $n{x}^{n-1}$ because $\frac{dx}{dx}=1$.

The same logic applies to more complex $y$-expressions:

• $\frac{d}{dx}(\sin y)=\cos y\cdot \frac{dy}{dx}$
• $\frac{d}{dx}(e^y)=e^y\cdot \frac{dy}{dx}$
• $\frac{d}{dx}(y^3)=3y^2\cdot \frac{dy}{dx}$

And for products involving both $x$ and $y$, the product rule applies:
$\frac{d}{dx}(xy)=y+x\frac{dy}{dx}$