Implicit Differentiation Calculator
Implicit differentiation is a technique for finding the derivative dy/dx when an equation defines y implicitly through both x and y rather than expressing y directly as a function of x alone. A classic example is the circle x squared plus y squared = 25: you cannot write y as a single function of x without splitting into two halves, so instead you differentiate both sides with respect to x, treating y as an implicit function of x and applying the chain rule where needed. The implicit function theorem gives a tidy formula: if you write the equation as F(x, y) = 0, then dy/dx = minus Fx divided by Fy, where Fx and Fy are the partial derivatives of F with respect to x and y. This calculator evaluates those partial derivatives numerically using the central difference method, which approximates Fx as (F(x+h, y) - F(x-h, y)) / (2h) for a small h, and Fy similarly. This means you can enter any equation expressible as F(x, y) = 0 using standard JavaScript math notation, including polynomial, trigonometric, exponential and logarithmic terms. Enter the expression for F(x, y), the x-coordinate and the y-coordinate of the point, and the calculator returns dy/dx at that point, the partial derivatives, and the equation of the tangent line. Use standard math notation: x*x or x**2 for squares, Math.sin(x), Math.exp(x), Math.log(x), and so on.
Example: x*x + y*y - 25 means x² + y² = 25. Use Math.sin(x), Math.exp(x), Math.log(x) etc.
dy/dx = -Fx / Fy. Partial derivatives are computed numerically by central difference. F(x,y) at the point should be near zero for valid on-curve points.
How it works
The calculator uses the implicit function theorem: for F(x, y) = 0, the slope is dy/dx = -Fx / Fy. The partial derivatives are approximated by the central difference formula: Fx ≈ (F(x+h, y) - F(x-h, y)) / (2h) and Fy ≈ (F(x, y+h) - F(x, y-h)) / (2h), where h = 1×10-7. The tangent line at (x0, y0) with slope m is y = m(x - x0) + y0. The F(x,y) value at the point shows how close the point is to the actual curve; values near zero mean the point lies on the curve.
Worked example
Equation: x² + y² - 25 = 0 (a circle of radius 5). Point: x = 3, y = 4. F(3, 4) = 9 + 16 - 25 = 0.0000 (on the curve). Fx = 2x = 6.0000, Fy = 2y = 8.0000. dy/dx = -6 / 8 = -0.7500. Tangent line: y = -0.75(x - 3) + 4 = y = -0.75x + 6.25. These match the defaults pre-filled above.
Related calculators
- Taylor Series Calculator: approximate functions with polynomial Taylor series.
- Newton's Method Calculator: find function roots using iterative Newton-Raphson steps.
- Trig Identity Calculator: verify trigonometric identities numerically at any angle.
- Percent Error Calculator: compare your computed slope to a reference value.