Move a point on a circle with the slider. Differentiate parametrically or differentiate implicitly, and you get the exact same tangent slope — because both methods are ultimately dealing with the same curve.
The equation of a circle, x²+y²=r², can't be neatly solved into the form y = f(x). An equation like this is called an implicit function. And yet there are two different ways to find the slope of its tangent line.
One is parametric differentiation. Write a point on the circle as x=r cos t, y=r sin t, and you can find the slope as dy/dx = (dy/dt)÷(dx/dt). The other is implicit differentiation. Differentiate both sides of x²+y²=r² with respect to x directly, but since y is itself a function of x, apply the chain rule to get 2x + 2y·y′ = 0, then solve for y′. Remarkably, the slope you get from both methods always comes out exactly the same — because in the end, you're finding the tangent line at the same point on the same curve either way. Move the point with the slider and check that the two values match.