Calculator Inputs
This solver handles structured implicit equations of the form F(x,y)=0 using polynomial, mixed, trigonometric, exponential, logarithmic, and constant terms. Trigonometric inputs are interpreted in radians.
Example Data Table
| Example equation | Point (x, y) | Computed dy/dx | Comment |
|---|---|---|---|
| x2 + y2 − xy − 4 = 0 | (2, 2) | −1 | Mixed term shifts the slope from a standard circle case. |
| x2 + y2 − 9 = 0 | (3/√2, 3/√2) | −1 | At equal coordinates on the circle, the tangent slope is negative one. |
| ln(x) + y2 − 1 = 0 | (1, 1) | −0.5 | Logarithmic terms require positive x-values. |
Formula Used
Implicit model: F(x,y)=0
Differentiate both sides with respect to x: Fx + Fy(dy/dx)=0
Solve for the derivative: dy/dx = −Fx / Fy
Partial derivative rules used by this calculator:
- d(Axn)/dx = A·n·xn−1
- d(Bym)/dx = B·m·ym−1·dy/dx
- d(Cxpyq)/dx = C[p·xp−1yq + xpq·yq−1·dy/dx]
- d(sin(y))/dx = cos(y)·dy/dx, d(cos(y))/dx = −sin(y)·dy/dx
- d(ey)/dx = ey·dy/dx, d(ln(y))/dx = (1/y)·dy/dx
How to Use This Calculator
- Enter coefficients for the terms you want in F(x,y)=0.
- Set exponents for xn, ym, and xpyq.
- Leave unused terms at zero to remove them from the equation.
- Choose an evaluation point (x0, y0).
- Set graph limits and grid density for the contour plot.
- Press the solve button to compute F(x0, y0), Fx, Fy, and dy/dx.
- Review the tangent line, curve status, and exported report if needed.
FAQs
1) What does this calculator solve?
It analyzes structured implicit equations written as F(x,y)=0. It computes partial derivatives, the implicit slope dy/dx, curve status at a chosen point, and a contour graph with a tangent line when possible.
2) Does it solve every possible implicit equation?
No. It is a template-based solver for common polynomial, mixed, trigonometric, exponential, logarithmic, and constant terms. It does not parse arbitrary free-form algebraic expressions typed as unrestricted text.
3) Why can dy/dx become undefined?
The derivative formula uses dy/dx = −Fx/Fy. If Fy is zero, the denominator vanishes. That usually signals a vertical tangent, a singular point, or the absence of a unique local slope.
4) What happens if my point is not on the curve?
The tool still evaluates the local derivative formula from the chosen coordinates, but it warns that the point does not satisfy F(x,y)=0. In that case, the tangent interpretation is only approximate.
5) Are trigonometric entries treated as degrees?
No. The calculator uses radians. That matches standard calculus differentiation rules and most scientific computing environments, so enter x and y values in radians whenever sine or cosine terms are active.
6) Why do logarithm inputs sometimes fail?
Natural logarithms require positive arguments. If ln(x) is active, x must stay positive at the evaluation point. If ln(y) is active, y must also stay positive, or the result becomes undefined.
7) Can I export the results?
Yes. After a successful calculation, you can download a CSV summary or generate a PDF report. Both exports include the point, equation details, slope data, tangent line, and calculation status.
8) What does the graph show?
The graph plots the contour F(x,y)=0 over your selected range. It also marks the chosen point and draws a tangent line when the point lies on the curve and the derivative is well-defined.