Implicit Differentiation

So far every curve has come solved for y: y = x^2, y = (3x+1)^5. But some curves tangle y up with x and refuse to be untangled. The circle of radius 5 is the classic example:

x^2 + y^2 = 25

We can still find the slope \frac{dy}{dx} — without ever solving for y. The trick is to differentiate both sides with respect to x, treating y as a (hidden) function of x. Then every y-term picks up an extra \frac{dy}{dx} by the chain rule:

\frac{d}{dx}\bigl[y^2\bigr] = 2y\cdot\frac{dy}{dx}

That is the chain rule with an outer function u^2 and an inner function y: bring the power down (2y), then multiply by the derivative of the inside (\frac{dy}{dx}). Differentiating the whole equation term by term:

2x + 2y\,\frac{dy}{dx} = 0

Now just solve for \frac{dy}{dx} — subtract 2x and divide by 2y:

\frac{dy}{dx} = -\frac{x}{y}

One formula gives the slope at every point of the circle. At (3,4) the slope is -\tfrac{3}{4}; at (4,3) it is -\tfrac{4}{3}. No solving for y required.

The recipe

To differentiate an equation that mixes x and y together:

Differentiate every term on both sides with respect to x.
For a term involving y, differentiate as usual and then multiply by \frac{dy}{dx} — that extra factor is the chain rule, since y is itself a function of x.
Collect all the \frac{dy}{dx} terms on one side, factor it out, and solve for \frac{dy}{dx}.

This is exactly how you find the slope of a curve — a circle, an ellipse, any relation — that isn't a function and so can never be written as y = f(x).

The slope of a circle, pictured

Here is x^2 + y^2 = 25 with the tangent line at (3,4). Implicit differentiation predicts a slope of -\frac{x}{y} = -\frac{3}{4} there — and the tangent line bears it out. Notice the tangent is perpendicular to the radius, just as it should be.