Implicit Differentiation#
Up to this point, we have always been writing our functions in the form
\[y = f(x)\]
to mean that the dependent variable \(y\) is a function of the independent variable \(x\). Another way to view \(y=f(x)\), is to think of it as an equation relating the two variables \(x\) and \(y\) where we have explicitly solved for \(y\) in terms of \(x\). In general, given an equation relating \(x\) and \(y\), it is not always going to be the case that \(y\) is isolated on one side of the equation, or that it is even possible to solve for \(y\) in terms of \(x\). But even in these cases, we can still think of \(y\) as a function of \(x\).