Implicit Derivative Calculator

The Implicit Derivative Calculator differentiates an equation in x and y with respect to x — without solving for y first — to the 1st, 2nd, or 3rd order. It shows every partial derivative used along the way, the resulting symbolic derivative, and an optional numeric evaluation at a point (x₀, y₀).

It's built for calculus students checking an implicit-differentiation homework problem, and for anyone who needs dy/dx (or a higher-order derivative) of a curve like a circle, an ellipse, or the folium of Descartes without isolating y by hand — which for most implicit equations isn't even possible with elementary functions.

How to use it

1. Enter the left side and right side of your equation in x and y — standard math notation works: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser. For example, a circle of radius 5 is x^2 + y^2 on the left and 25 on the right.
2. Choose the order — 1st derivative (dy/dx), 2nd derivative (d²y/dx²), or 3rd derivative (d³y/dx³).
3. Optionally enter a point (x₀, y₀) to also get a numeric value — leave both blank for the symbolic result only.
4. Read Implicit Derivative for the symbolic result and Step-by-Step Derivation for every partial derivative used to get there.

Results update automatically as you type. For example, the circle x^2 + y^2 = 25 at (3, 4) returns dy/dx = -(x / y) = -0.75.

How does implicit differentiation work?

Writing the equation as H(x, y) = 0 (left side minus right side), differentiating H(x, y(x)) = 0 with respect to x — treating y as a function of x via the chain rule — and solving for the derivative gives:

dy/dx = -Hx / Hy

where Hx and Hy are the ordinary partial derivatives of H, each computed by treating the other variable as a constant. This sidesteps ever solving the original equation for y, which for a curve like the folium of Descartes (x³ + y³ = 9xy) isn't possible in closed form at all.

The 2nd and 3rd derivatives repeat the same trick on the resulting relation, substituting the lower-order result back in each time:

d²y/dx²  = -(Hxx + 2·Hxy·y′ + Hyy·y′²) / Hy
d³y/dx³  = -(Hxxx + 3·Hxxy·y′ + 3·Hxyy·y′² + Hyyy·y′³ + 3·Hxy·y″ + 3·Hyy·y′·y″) / Hy

What if the answer contains both x and y?

That's expected — unlike an explicit derivative, an implicit derivative is a function of both x and y, since y itself isn't isolated. For the circle example above, dy/dx = -(x / y) is only a number once you also fix a point on the curve (like (3, 4)), which is exactly what the optional x₀/y₀ fields are for.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2*y - 2*x), trigonometric functions (sin(x*y), cos(y)), roots and logarithms (sqrt(x), log(y)), and combinations of these, using x and y as the two variables. Angles in trig functions are in radians by default — write pi for π.

Working with a function that's already explicit, f(x, y, z), and need its directional derivative or gradient instead? See the Directional Derivative Calculator. To evaluate arbitrary math expressions directly, see the Math Evaluator.

Privacy

This calculator runs entirely in your browser. Your equation, order, and point are never uploaded, logged, or stored — the computation happens locally on your device.