Implicit Differentiation Calculator

Q: What is implicit differentiation?

A way to find \frac{dy}{dx} when y is not isolated, as in x^2 + y^2 = 25. Differentiate both sides with respect to x, treating y as a function of x (so every y term picks up a factor \frac{dy}{dx} by the chain rule), then solve for \frac{dy}{dx}.

Q: Why does d/dx of y² give 2y·dy/dx?

Because y depends on x. By the chain rule, \frac{d}{dx}[y^2] = 2y\cdot\frac{dy}{dx}, just as \frac{d}{dx}[(g(x))^2] = 2g(x)\,g'(x).

Q: What is the shortcut formula?

Move everything to one side, F(x,y) = 0. Then \frac{dy}{dx} = -\frac{F_x}{F_y}, using partial derivatives. This calculator uses it and shows both partials.

Q: How do I find the tangent line to an implicit curve?

Plug the point into \frac{dy}{dx} to get the slope, then use point-slope form. Enter a point on the curve below and the tangent is drawn.

Get dy/dx for curves you can’t solve for y, like circles, ellipses and folium curves, with the steps and the curve drawn with its tangent at your point.

Equation in x and y

Try:

Point x₀y₀

Answer

\frac{dy}{dx} = -\frac{x}{y}

At $(3, 4)$ : slope $-\frac{3}{4}$ , tangent $y = -\frac{3}{4}x + \frac{25}{4}$

Step by step

Move everything to one side
$F(x,y) = x^{2} + y^{2} - 25 = 0$
Differentiate with respect to x
Treat y as constant for F_x.
$F_x = 2 x$
Differentiate with respect to y
Treat x as constant for F_y.
$F_y = 2 y$
Solve for dy/dx
Differentiating F(x, y(x)) = 0 with the chain rule gives F_x + F_y·y′ = 0, so y′ = −F_x / F_y.
$\frac{dy}{dx} = -\frac{F_x}{F_y} = -\frac{x}{y}$

Still stuck? Ask the tutor

An AI tutor reads the problem and steps above and explains them in plain words. It can make mistakes, so check it against the worked steps.

Doing it by hand

Take $x^2 + y^2 = 25$ . Differentiate each term with respect to $x$ : $2x + 2y\,\frac{dy}{dx} = 0$ . The $\frac{dy}{dx}$ appears because $y$ is a function of $x$ (the chain rule). Solve: $\frac{dy}{dx} = -\frac{x}{y}$ . At $(3,4)$ the slope is $-\tfrac34$ , perpendicular to the radius, as geometry says it must be.

The calculator uses the equivalent shortcut $\frac{dy}{dx} = -F_x/F_y$ with partial derivatives, which gives the same answer for any equation.

Questions students ask

What is implicit differentiation?

A way to find $\frac{dy}{dx}$ when $y$ is not isolated, as in $x^2 + y^2 = 25$ . Differentiate both sides with respect to $x$ , treating $y$ as a function of $x$ (so every $y$ term picks up a factor $\frac{dy}{dx}$ by the chain rule), then solve for $\frac{dy}{dx}$ .

Why does d/dx of y² give 2y·dy/dx?

Because $y$ depends on $x$ . By the chain rule, $\frac{d}{dx}[y^2] = 2y\cdot\frac{dy}{dx}$ , just as $\frac{d}{dx}[(g(x))^2] = 2g(x)\,g'(x)$ .

What is the shortcut formula?

Move everything to one side, $F(x,y) = 0$ . Then $\frac{dy}{dx} = -\frac{F_x}{F_y}$ , using partial derivatives. This calculator uses it and shows both partials.

How do I find the tangent line to an implicit curve?

Plug the point into $\frac{dy}{dx}$ to get the slope, then use point-slope form. Enter a point on the curve below and the tangent is drawn.