Implicit differentiation

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

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

Using the chain rule:

${\displaystyle {\begin{aligned}{\tfrac {df}{dx}}&={\tfrac {df}{dy}}\times {\tfrac {dy}{dx}}\\&=2y\times {\tfrac {dy}{dx}}\end{aligned}}}$

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

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

Working:

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