Finding dy/dx for x² + y² = 1

Real Analysis · Axiom Academy

EXAMPLE Finding for x^2 + y^2 = 1 Differentiating the unit circle implicitly to get the tangent slope at any point. The unit circle x^2 + y^2 = 1 does not define y as a single function of x — for most x there are two values of y . Use implicit differentiation to find , then read off the tangent slope at a few points on the circle. At any point (x,y) the tangent slope is — exactly perpendicular to the radius (whose slope is ), since . At the point the radius has slope 1 and the tangent has slope -1 . Check the formula at three points Plug each point into and confirm the slope matches the picture. Nice work — you differentiated a curve that can't be solved for y , and got a slope formula that works everywhere on the circle. Implicit differentiation lets you differentiate x^2 + y^2 = 1 without first solving for y . The chain rule is the whole trick: y depends on x , so , not just 2y . The result gives the tangent slope at any point (x,y) on the circle. It matches the geometry: the tangent is perpendicular to the radius, since — horizontal at (0,1) , vertical at (1,0) . A zero denominator flags a vertical tangent: when y = 0 the slope is undefined, exactly where the circle has a vertical tangent line. The same move works for any implicitly defined curve: differentiate term by term and apply the chain rule every time you hit a y .

This is the written version of the interactive lesson above. See the full Real Analysis course.