Enter partial derivatives, coordinates, and optional second derivatives. Get slope, tangent, normal, and exports instantly. Practice implicit differentiation with clear steps and flexible inputs.
Enter evaluated partial derivatives at the chosen point. The relation label is stored for reference and exports.
| Time | Relation | Point | Fx | Fy | dy/dx | d²y/dx² | Status |
|---|---|---|---|---|---|---|---|
| No calculations saved yet. | |||||||
| Relation | Point | Fx | Fy | Fxx | Fxy | Fyy | dy/dx |
|---|---|---|---|---|---|---|---|
| x² + y² - 25 = 0 | (3, 4) | 6 | 8 | 2 | 0 | 2 | -0.75 |
| x² + xy + y² - 7 = 0 | (1, 2) | 4 | 5 | 2 | 1 | 2 | -0.8 |
| x³ + y³ - 6xy = 0 | (3, 3) | 9 | 9 | 12 | -6 | 12 | -1 |
| x² - y = 0 | (2, 4) | 4 | -1 | 2 | 0 | 0 | 4 |
For an implicit relation F(x, y) = 0, the first derivative is found by differentiating both sides with respect to x:
dy/dx = -Fx / Fy
Here, Fx is the partial derivative of F with respect to x, and Fy is the partial derivative of F with respect to y.
If second partial derivatives are available, the second derivative is:
d²y/dx² = -(Fxx + 2Fxy(dy/dx) + Fyy(dy/dx)²) / Fy
The tangent line through (x₀, y₀) uses point-slope form. The normal line is perpendicular to the tangent.
Implicit differentiation is used when y is not isolated. Many algebra, calculus, and analytic geometry problems use this format. Circles, ellipses, mixed polynomial curves, and constraint equations often need this method. A fast calculator helps students verify slopes and line equations without repeating every algebra step by hand.
This page computes the first derivative from partial derivative values at a selected point. It also finds the tangent line, the normal line, and the tangent angle when the slope exists. If you provide second partial derivatives, it also returns the second derivative. That gives a stronger local picture of the curve near the chosen point.
This version focuses on reliable numeric evaluation. Instead of parsing a raw symbolic equation, it accepts Fx and Fy already evaluated at the point. That makes the tool simpler to maintain and easier to audit. It is also practical for class notes, exam review, engineering checks, and worked examples where partial derivatives are already known.
The page stores recent calculations in a history table. You can export that history to CSV for revision or worksheet creation. You can also save the current output as a PDF. The Plotly graph provides a quick local view of the chosen point together with tangent and normal directions. This helps learners connect symbolic differentiation to geometric meaning.
You need x, y, Fx, and Fy at the selected point. Those values are enough to compute dy/dx, plus tangent and normal line information.
This version does not parse symbolic equations. It expects the needed partial derivatives already evaluated at the chosen point.
If Fy is zero and Fx is nonzero, dy/dx is undefined at that point. The calculator treats that case as a vertical tangent.
Then the slope is indeterminate under the basic rule. That usually means the point is singular or needs deeper local analysis.
Those optional values let the calculator estimate the second derivative. That helps describe local curvature and concavity behavior.
No. The graph shows the selected point together with tangent and normal views. It is a local visualization, not a full symbolic curve plot.
Yes. The history table stores recent calculations in the session. Use the CSV button to export the saved history rows.
Yes. It is useful for checking answers, reviewing line equations, and comparing several worked examples quickly.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.