Implicit Differentiation
Simple Explanation
When an equation relates x and y together, but y cannot easily be isolated as y=(some function of x), differentiate BOTH sides of the equation with respect to x directly — treating y as an unknown function of x. Every time a y-term is differentiated, the Chain Rule adds an extra dy/dx factor.
Why Do We Need It?
Many important curves (like circles, ellipses, and other implicit relations) simply cannot be written as a single y=f(x) — implicit differentiation is the only way to find their tangent slopes directly.
See It
A circle of radius 5 centered at the origin, with a radius line to the point (3,4) and a tangent line touching the circle at that point
Formula
Implicit Differentiation
Differentiate both sides w.r.t. x; for every y-term, multiply by dy/dx (Chain Rule)
When y is not isolated on one side of an equation, differentiate both sides of the equation with respect to x directly, treating y as an unknown function of x — every time a y appears, its derivative contributes a dy/dx factor by the Chain Rule.
- dy/dx
- — the derivative of y with respect to x — the very thing being solved for
When to use it: Whenever an equation relates x and y together, but y cannot easily be isolated as y = (some function of x).
Worked Example
Find a tangent slope using implicit differentiation
Given the circle x² + y² = 25, find dy/dx (the slope of the tangent line) at the point (3, 4).
Why Does This Work?
Since y is (implicitly) a function of x along the curve, y² is really a composite function — "square of y(x)" — so the Chain Rule genuinely applies: d/dx[y²] = 2y·(dy/dx), exactly as d/dx[u²]=2u·(du/dx) for any function u(x). This is the same Chain Rule from section 11.3, just applied to y instead of a named inner function.
Real-Life Example
Analyzing a curved mechanical cam profile
An engineer designs a curved cam or gear profile described by an implicit equation relating x and y, where y cannot be isolated as a simple function of x.
Implicit differentiation gives the slope of the profile at any point directly from the defining equation, without needing to solve for y first.
Practice
For the circle x² + y² = 25, find dy/dx at the point (0, 5).
HardCommon mistake
Forgetting to apply the Chain Rule to y-terms — writing d/dx[y²] = 2y instead of the correct 2y·(dy/dx), since y is implicitly a function of x, not an independent variable like x itself.
Quick Review
- Differentiate both sides of the equation with respect to x.
- Every y-term picks up an extra factor of dy/dx (Chain Rule).
- Solve algebraically for dy/dx afterward.