Implicit differentiation: for F(x,y) = 0, differentiate both sides with respect to x, treating y as y(x). Then dy/dx = −Fₓ/Fᵧ. Works when the curve cannot be solved for y explicitly (circles, ellipses, folium of Descartes).
How to Use
Select a curve:
- F(x,y) = 0: Implicit equation
- (x₀,y₀): Point on curve
- Output: dy/dx at point
Method
1) Differentiate term by term. 2) For yⁿ terms: d/dx[yⁿ] = nyⁿ⁻¹·(dy/dx). 3) Collect dy/dx terms on one side. 4) Solve for dy/dx.
Second Derivative
d²y/dx² = d/dx[dy/dx]. Differentiate the dy/dx expression again (implicitly), substituting dy/dx back in. Result is d²y/dx² in terms of x and y.
Step-by-Step Instructions
- 1Select implicit curve.
- 2Enter point (x₀,y₀).
- 3Get dy/dx formula.
- 4Evaluate at point.
- 5Find tangent slope.