Implicit Differentiation Calculator

Implicit differentiation finds dy/dx when y is not given explicitly as a function of x, but is instead defined by an equation that mixes both: F(x, y) = 0.

The trick: differentiate both sides with respect to x, treating y as a function of x — so y² becomes 2y·(dy/dx) by the chain rule, sin(y) becomes cos(y)·(dy/dx), and so on. Then solve for dy/dx.

The shortcut. For F(x, y) = 0, a clean closed form falls out:

dy/dx = − F_x / F_y

where F_x = ∂F/∂x and F_y = ∂F/∂y are the partial derivatives. The minus sign comes from rearranging the total differential dF = F_x · dx + F_y · dy = 0. This is the formula the calculator uses.

What the calculator handles. A general conic Ax² + By² + Cxy + Dx + Ey + F = 0 — which covers circles, ellipses, hyperbolas, parabolas, and degenerate pairs of lines. The partials work out cleanly:

F_x = 2Ax + Cy + D F_y = 2By + Cx + E

Worked example — the unit circle x² + y² = 25 at (3, 4). Coefficients: A = 1, B = 1, C = 0, D = 0, E = 0, F = −25. Then F_x = 2x = 6 and F_y = 2y = 8. So dy/dx = −6/8 = −3/4.

Sanity check: at (3, 4) the radius vector points in direction (3, 4), so the tangent line is perpendicular to it. The radius has slope 4/3, so a perpendicular line has slope −3/4. ✓

A harder example — folium of Descartes, x³ + y³ = 6xy. Differentiate both sides with respect to x:

3x² + 3y² · dy/dx = 6y + 6x · dy/dx

Group the dy/dx terms on one side, factor, and divide:

dy/dx = (6y − 3x²) / (3y² − 6x) = (2y − x²) / (y² − 2x)

You couldn’t easily solve the original equation for y as a function of x — the cubic has three branches — but implicit differentiation gives the slope at any point without that.

One more — finding the tangent line at a point. For x² + xy + y² = 7 at (1, 2): differentiate to get 2x + y + x·(dy/dx) + 2y·(dy/dx) = 0. Plug in x = 1, y = 2: 2 + 2 + 1·(dy/dx) + 4·(dy/dx) = 0, which solves to dy/dx = −4/5. The tangent at (1, 2) has slope −4/5.

Multi-valued curves. Implicit differentiation handles cases where y is not a single-valued function of x. The full circle x² + y² = 25 is multi-valued — every x in (−5, 5) corresponds to two y values, top and bottom. Implicit differentiation gives the correct local slope at each specific point regardless of which branch it lives on.

When F_y = 0 — vertical tangents. If F_y vanishes at the point of interest, the curve has a vertical tangent there and dy/dx is undefined. The calculator flags this case and reports dx/dy instead, which is well-defined.