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}$$