# HSC Extension 1 and 2 Mathematics/3-Unit/Preliminary/Harder 2-Unit

## §Implicit differentiation

Implicit differentiation is a method of differentiating an expression in $x$ and $y$, where $x$ and $y$ are related in some manner and neither are constant.

For example, one could differentiate $f(x) = y^2$ with respect to $x$ as follows:

Using the chain rule:
\begin{align} \tfrac{df}{dx} & = \tfrac{df}{dy} \times \tfrac{dy}{dx} \\ & = 2y \times \tfrac{dy}{dx} \end{align}

It is useful to think of implicit differentiation as normal differentiation with respect to $x$, only whenever you come across a term with $y$, you multiply the differentiated term by $dy/dx$.

Another example: find the derivative $dy/dx$ of $x^2+y^2=r^2$

Working:

\begin{align} 2x+2y.\frac{dy}{dx} & = 0 \\ 2x & = -2y.\frac{dy}{dx} \\ \frac{dy}{dx} & = -\frac{x}{y} \end{align}