What is implicit differentiation?

Implicit differentiation finds dy/dx when y is not isolated as an explicit function of x, but both x and y appear together in an equation such as x squared plus y squared equals 25. By differentiating both sides with respect to x and treating y as a function of x, you can solve for dy/dx using the implicit function theorem: dy/dx = minus Fx divided by Fy, where Fx and Fy are the partial derivatives of F with respect to x and y.

How do you find dy/dx implicitly for x squared plus y squared equals 25?

Write F(x,y) = x squared plus y squared minus 25 = 0. Then Fx = 2x and Fy = 2y. So dy/dx = minus Fx divided by Fy = minus 2x divided by 2y = minus x divided by y. At the point (3, 4), dy/dx = minus 3 divided by 4 = minus 0.75.

What is the tangent line at a point found by implicit differentiation?

Once you have the slope dy/dx at a point (x0, y0), the tangent line is y minus y0 = m times (x minus x0), where m = dy/dx at that point. For x squared plus y squared = 25 at (3, 4), the slope is minus 0.75 and the tangent line is y minus 4 = minus 0.75 times (x minus 3), which simplifies to y = minus 0.75x + 6.25.

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.

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F(x, y) expression (set equal to 0)

Example: x*x + y*y - 25 means x² + y² = 25. Use Math.sin(x), Math.exp(x), Math.log(x) etc.

x value

y value

-0.7500

dy/dx at the given point

Fx (partial x)6.0000

Fy (partial y)8.0000

F(x,y) at point0.0000

Tangent liney = -0.75x + 6.25

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.

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