Curves defined by an equation

See an equation in x and y as a curve, and find tangents without solving for y. About 8 minutes, A-level Further Maths.

What you will do

1. Every point that fits. The curve is every point where F(x, y) = x^2 + y^2 equals the level c = 4 — a circle of radius 2. You do not need to solve for y to draw it: the engine checks the equation across the plane and joins the points that satisfy it.
2. Change the level. Move the level slider. Each value of c gives a different circle: x^2 + y^2 = c. Make the circle radius about 3 (so c 9).
3. The gradient. The orange arrow is the gradient (F_x,\ F_y) = (2x,\ 2y). It points straight away from the curve, so the tangent line is perpendicular to it.
4. Slope without solving. The tangent's slope is dydx = -F_xF_y. On x^2 + y^2 = 25 at the point (3, 4):
5. A different curve. Now F = xy with c = 1: a hyperbola. Drag the point along it and watch the slope change sign between the two branches.
6. Your turn. Everything is unlocked. Type any F(x, y) — try x + y or (x^2+y^2-1)^3 - x^2y^3 — move the level, and drag the point.