The equation of a circle, x²+y²=r², can't be neatly solved for y as an expression in x alone (you can with a square root, but that splits it into two equations, one for the top half and one for the bottom half). An equation that isn't explicitly rearranged into the form y=f(x) like this is called an implicit function. And yet there's a way to find the tangent slope of an implicit function like this — actually, two ways.
The first method is parametric differentiation. Write a point on the circle using an angle t, and you get x=r cos t, y=r sin t. When both x and y are expressed in terms of a third variable, t, like this, you find dy/dx as the ratio of dy/dt to dx/dt, each differentiated with respect to t: dy/dx = (dy/dt)÷(dx/dt). For a circle, dx/dt=−r sin t and dy/dt=r cos t, so dy/dx = (r cos t)÷(−r sin t) = −cos t/sin t.
The second method is implicit differentiation. Just differentiate both sides of x²+y²=r² with respect to x. Differentiating x² still gives 2x, same as always, but differentiating y² takes some care. y is actually a function of x (even though it isn't solved out as an expression), so applying the chain rule to differentiate y² gives 2y·(dy/dx). The r² on the right is a constant, so it differentiates to 0. That leaves you with 2x + 2y·y′ = 0, and solving that for y′ gives y′ = −x/y.
Here's where something interesting shows up. Is −cos t/sin t, the value from parametric differentiation, really the same as −x/y, the value from implicit differentiation? Substitute x=r cos t, y=r sin t in, and −x/y = −(r cos t)/(r sin t) = −cos t/sin t — the two expressions match exactly! Getting the same answer from two completely different approaches isn't a coincidence. Both methods are ultimately finding the tangent line at "the same point on the same circle," so even though the way they're expressed differs, the answer has no choice but to come out as one single value.
When do these two differentiation methods actually get used? Parametric differentiation comes up naturally in situations where both x and y are determined by time t — the path of an object moving over time (say, a projectile's trajectory, or a point on a spinning wheel). Implicit differentiation is handy for finding tangent lines on equations like circles and ellipses that are hard to neatly solve into y=f(x) form. Knowing both means you can pick whichever one is easier for the situation at hand.
On our activity page, you can adjust a point on a circle by angle with a slider, and watch the calculation unfold step by step in both the parametric differentiation tab and the implicit differentiation tab. Switch back and forth between the two tabs and compare, right down to the decimal, whether the same point really gives the same slope.