Implicit Differentiation

Real Analysis · Axiom Academy

LESSON Implicit Differentiation Finding the slope of a curve when y is tangled up with x and can't be isolated — differentiate both sides and solve. An explicit relation hands you y alone on one side, so each x has exactly one y and the slope is routine. An implicit relation mixes x and y together. Sweep a vertical line across each curve below: the parabola is hit once at every x , but the circle is hit twice — that second intersection is exactly why no y = f(x) exists, and why we need a new tool. The whole method is four moves. The one that trips people up is the second: because y secretly depends on x , differentiating a y -term fires the chain rule , attaching a factor of y' . Watch x nudge forward by dx below — y responds by , and y^2 responds by . The y' rides along on every y . Differentiate both sides with respect to x . Each y -term carries the chain rule — attach a y' (since y is a function of x ). Collect every y' term on one side. the power-and-chain rule on a y -term a product like xy needs the product rule too Differentiating a plain power of x gives . Differentiating the same power of y gives — identical, plus the chain-rule y' . Apply the procedure to x^2 + y^2 = 25 . Differentiating gives — note the y' riding the y^2 term — and solving leaves a slope that depends on where you are. Below, a point orbits the circle; the tangent stays glued to it, and the readout computes -x/y live from the point's coordinates.

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