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:

$\begin{array}{cc}\frac{df}{dx}& =\frac{df}{dy}\times \frac{dy}{dx}\\ & =2y\times \frac{dy}{dx}\end{array}$

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:

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