Implicit Differentiation Calculator

Work out implicit differentiation instantly with clear inputs, formula shown and shareable results.

About Implicit Differentiation Calculator

Differentiating a·x² + b·y² = c with respect to x treats y as a function of x, giving 2ax + 2by·(dy/dx) = 0 and hence dy/dx = −ax/(by). For a = b = 1 this is the circle, where the tangent is perpendicular to the radius.

Formula

Implicit derivative

a·x² + b·y² = c ⇒ dy/dx = −ax / (by)

Worked example

With the default values below, Implicit Differentiation Calculator returns dy/dx at the point of -0.75000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Coefficient a in a·x² + b·y² = c: 1
Coefficient b: 1
Point x: 3
Point y: 4

Results

dy/dx at the point: -0.75000000
Tangent line: y - 4 = -0.75(x - 3)
Value of a·x² + b·y²: 25.000000
Normal slope: 1.33333333

Step by step

1Values used
2Implicit derivative
3dy/dx at the point= -0.75000000
4Tangent line= y - 4 = -0.75(x - 3)
5Value of a·x² + b·y²= 25.000000
6Normal slope= 1.33333333

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For the circle x² + y² = 25 at (3, 4)?

dy/dx = −3/4, and the normal slope 4/3 points along the radius, as it must.

Why use implicit differentiation?

Because solving for y first may be impossible or messy, as with x³ + y³ = 6xy.