Your result
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Enter your values, then click Calculate result.How this calculator helps
Differentiate a supported polynomial equation in x and y, show both partial derivatives and check the slope at a supplied point. Enter a sum of polynomial monomials in x and y, such as x^2+y^2=25 or x*y+x=4. Use a caret for a non-negative integer exponent and an asterisk when multiplying terms. Numeric decimal coefficients are allowed. Parentheses, division, functions such as sin or log, negative powers and scientific notation are not supported. Rewrite a supported polynomial into an expanded sum before entry. The parser rejects other syntax instead of evaluating arbitrary code.
How to use it
- 1
Enter a polynomial equation using x, y, signed terms and non-negative integer powers, such as x^2+y^2=25.
- 2
Enter the point coordinates to evaluate the partial derivatives and test whether the point lies on the curve.
- 3
Click Calculate result to submit the inputs. Editing fields keeps the last submitted output until you click again.
- 4
Check the worked example, units and method limits before using the result.
Formula and methodology
For F(x,y)=0, F_x + F_y × dy/dx = 0, so dy/dx = −F_x/F_y where F_y≠0. Differentiate each supported monomial by its x or y exponent.
The calculator applies the disclosed heuristic to values entered on this device. It does not load private admissions data or claim to reproduce an institution's review process.
Worked calculation example
For x²+y²=25, dy/dx=−2x/(2y). At (3,4), the curve residual is zero and the slope is −0.75.
How to interpret your result
Expanded polynomial sums in x and y, total degree at most twelve, bounded coefficients and coordinates only. No general function parser, equation solving, branch selection or singular-point classification. The examples identify the method and input basis, so the result can be checked without guessing a hidden convention.
For different inputs or formulas, use Partial Fraction Calculator; Decimals Calculator; Surface Area Calculator.
Related questions this calculator covers
- implicit differentiation calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| x²+y²=25 at (3,4) | slope −0.75. |
| Same circle at (5,0) | F_y=0, so the finite formula is undefined. |
| Same circle at (1,1) | residual −23; point is not on the curve. |
Common mistakes to avoid
- Entering an unsupported function and assuming it was interpreted.
- Calling a partial-derivative ratio at an off-curve point the curve’s slope.
- Assuming a zero denominator alone classifies a singular point.
Substitute the same inputs into the disclosed equation and compare with this example: For x²+y²=25, dy/dx=−2x/(2y). At (3,4), the curve residual is zero and the slope is −0.75. Keep the method, units and source assumptions with any recorded result.
Authoritative reference. Expanded polynomial sums in x and y, total degree at most twelve, bounded coefficients and coordinates only. No general function parser, equation solving, branch selection or singular-point classification.What can affect the result?
Moving both sides into F
The calculator treats the equation as F(x,y)=left side minus right side. If no equals sign is present, the entered expression is understood as equal to zero. It then forms partial derivatives by holding the other variable constant. For an x-dependent monomial, the x derivative multiplies its coefficient by the x exponent and reduces that exponent by one. The y derivative follows the same rule. The symbolic output leaves algebraic simplification visible rather than pretending to solve y explicitly.
Derivative formula and point evaluation
The implicit derivative is negative F_x divided by F_y. A finite slope from that expression requires a nonzero F_y at the point. The point fields do not solve for an unknown coordinate; they evaluate the point you supply. The curve residual shows whether those coordinates actually satisfy the entered equation within a numerical tolerance. If they do not, the output refuses to describe the ratio as the curve’s slope at that point.
Singular points and method limits
When F_y is zero, the displayed derivative formula is undefined. That can correspond to a vertical tangent or a more complicated singular case; this tool does not choose an interpretation automatically. Numeric point evaluations use floating-point arithmetic and finite input limits. The supported polynomial degree is capped at twelve to keep parsing and evaluation bounded. The page is a polynomial differentiation aid, not a general computer algebra system for every implicit expression.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
Expanded polynomial sums in x and y, total degree at most twelve, bounded coefficients and coordinates only. No general function parser, equation solving, branch selection or singular-point classification.
Sources and review information
Frequently asked questions
Can I differentiate without solving for y?＋
Yes. Implicit differentiation treats y as dependent on x and combines partial derivatives into −F_x/F_y. This tool displays that symbolic relationship for the supported polynomial equation, avoiding an unnecessary branch choice from explicitly solving for y.
What should I enter for a circle?＋
For a circle centered at the origin with radius five, enter x^2+y^2=25. Supply a point on that circle if you want a slope check. The point (3,4) has zero residual and gives −0.75, whereas an off-curve point is identified separately.
Does it accept trigonometric equations?＋
No. The supported input is an expanded polynomial sum. Functions, parentheses and division require a larger expression system and are rejected here. The title and method disclosure make this restriction explicit rather than claiming general symbolic support.
Why does it say the point is not on the curve?＋
Substituting the supplied coordinates into left side minus right side produces a residual beyond the numerical tolerance. The partial-derivative ratio at an unrelated point is not the slope of that curve there. Check the equation and coordinates before calculating again.
Does F_y=0 always mean a vertical tangent?＋
Not always. It means this finite derivative formula cannot be used at the point. A vertical tangent or singular behavior needs further analysis, especially if both partial derivatives vanish. The calculator reports the limitation rather than selecting a geometric conclusion from one zero value.