What “Implicit” Means and Why It Matters

The relationships in What "Implicit" Means and Why It Matters may define a curve without giving one global formula $y=f(x)$. Along that curve, however, $y$ can still respond locally to changes in $x$. Implicit differentiation captures that local dependence by differentiating every term and attaching $dy/dx$ whenever a term containing $y$ is differentiated.

Keep the point of interest visible. The resulting derivative usually depends on both $x$ and $y$, so a numerical slope requires both coordinates. Also remember that a relation may have vertical tangents or multiple branches; the local derivative is meaningful even when a single global explicit formula is not.

An explicit equation says "here is $y$ in terms of $x$." An implicit equation says "here is a condition that $x$ and $y$ satisfy together." Circles, ellipses, and many physical constraints are more naturally written in this second form.

The curve may still behave like a function locally even when no single formula covers the entire shape. Implicit differentiation extracts local slope information directly from the relationship.