Implicit Differentiation Calculator with Steps

One box, pre-filled with x^2+y^2-25. Click Calculate. The implicit differentiation calculator form takes the left side of your equation and returns two rows: dy/dx, the slope, and the second derivative d²y/dx².

• Move the right side over first. The implicit differentiation calculator box takes F(x, y) = 0, everything on the left. From x^2+y^2 = 25, type x^2+y^2-25.
• Read the dy/dx row. The implicit differentiation calculator renders the slope as a formula: x^2+y^2-25 comes back as -x/y.
• Read the second derivative row. The implicit differentiation calculator differentiates again: x^2+y^2-25 comes back as -(x² + y²)/y².
• Try the example buttons. Each button drops a ready equation into the implicit differentiation calculator. x^3+y^3-6*x*y returns (2y - x^2)/(y^2 - 2x). The trig mix sin(x)+y^2-cos(x) works the same.
• Turn the formula into a number. Substitute a point: the implicit differentiation calculator answer -x/y reads -3/4 at (3, 4).

This page is more than a static implicit differentiation calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. "Find dy/dx for x^2+y^2 = 25 and the tangent slope at (3, 4)." The AI fills the box and runs the implicit differentiation calculator for you.
• Take a photo of your homework. No typing: the AI reads the equation and fills the implicit differentiation calculator form.
• Ask about the ideas, not just the number. "Why does every y term get a dy/dx?" The AI explains step by step — every number still comes from the implicit differentiation calculator on this page.
• Work backwards, then keep asking. Give a slope and a point, and ask which curves share them. Follow-ups are welcome — the implicit differentiation calculator page keeps teaching.

Picture the circle $x^2+y^2 = 25$ — radius 5, centered at the origin. Stand at the point $(3, 4)$. What is the slope under your feet?

In words first: for this circle the slope is negative x over y. In symbols, the fact every implicit differentiation calculator builds on is

$\dfrac{dy}{dx} = -\dfrac{x}{y}$

Now substitute the point: $dy/dx = -3/4$. The tangent at $(3, 4)$ drops three units for every four across, as the picture shows.

Why does an implicit differentiation calculator exist at all? Solving for y gives $y = \pm\sqrt{25 - x^2}$ — two functions glued into one curve. Love's 1916 calculus puts it plainly: x and y are "connected by an equation not solved for y." An implicit differentiation calculator skips the solving: it treats y as a function of x and differentiates the equation as it stands. That one move is what every implicit differentiation calculator automates — Example 2's folium never solves for y at all.

Four moves power every implicit derivative calculator result:

1. Differentiate both sides with respect to x. Every y term carries the chain rule: $y^2$ becomes $2y \cdot dy/dx$.
2. Move the dy/dx terms left, everything else right.
3. Factor dy/dx out on the left.
4. Divide to solve for dy/dx, the result row of the implicit differentiation calculator.

In symbols the four moves collapse into one line — the formula this implicit differentiation calculator runs:

$\dfrac{dy}{dx} = -\dfrac{\partial F/\partial x}{\partial F/\partial y}$

The form computes diff(F, x) and diff(F, y) exactly as written. The implicit differentiation calculator row matches hand work step for step.

The classic first job for an implicit differentiation calculator: the circle $x^2+y^2 = 25$, pre-loaded in the form above.

1. Equation as F(x, y) = 0: $x^2+y^2-25 = 0$ — the exact string the implicit differentiation calculator expects.
2. Differentiate both sides: $2x + 2y \cdot dy/dx = 0$ — the chain rule tags $y^2$ with $dy/dx$.
3. Isolate and divide: $2y \cdot dy/dx = -2x$, so $dy/dx = -x/y$ — the exact row the implicit differentiation calculator prints.
4. Slope at (3, 4): $-3/4$, so the tangent line is $y = -\frac{3}{4}x + \frac{25}{4}$.

Notice the answer still contains y — the implicit differentiation calculator leaves it that way on purpose. An implicit derivative is a formula in x and y; points turn it into numbers. The radius to $(3, 4)$ has slope $4/3$; the tangent slope $-3/4$ is its negative reciprocal, exactly as the picture shows.

Type x^2+y^2-25 into the implicit differentiation calculator above — the dy/dx row reads $-x/y$.

A loopier curve for the implicit differentiation calculator: the folium of Descartes $x^3+y^3 = 6xy$. The loop in the first quadrant is an implicit differentiation solver favorite.

1. Move everything left: $x^3+y^3-6xy = 0$ — one of the implicit differentiation calculator's example buttons.
2. Differentiate: $3x^2 + 3y^2 \cdot dy/dx - 6y - 6x \cdot dy/dx = 0$ — product rule on $6xy$, chain rule on $y^3$.
3. Factor: $dy/dx \cdot (3y^2 - 6x) = 6y - 3x^2$.
4. Divide and cancel the 3: $dy/dx = \dfrac{2y - x^2}{y^2 - 2x}$.
5. Slope at (3, 3): $\dfrac{6 - 9}{9 - 6} = -1$ — the point sits on the loop since $27 + 27 - 54 = 0$.

Enter x^3+y^3-6*x*y in the implicit differentiation calculator above. The dy/dx row reads exactly $\frac{2y-x^2}{y^2-2x}$ — no solving for y anywhere. That is the implicit differentiation calculator handling a curve no explicit formula can describe.

Three slips the implicit differentiation calculator catches every day:

1. Dropping the chain rule on y. The derivative of $y^2$ is $2y \cdot dy/dx$, never just $2y$. Every y term needs its dy/dx tag — the number-one slip an implicit differentiation solver sees. The implicit differentiation calculator never drops it.
2. Expecting an x-only answer. An implicit derivative usually still contains y — the implicit differentiation calculator keeps it that way. Substitute a point instead: -x/y at (3, 4) gives -3/4.
3. Leaving the right side in. The box takes F(x, y) = 0, so from x^2+y^2 = 25 type x^2+y^2-25. Anything left on the right side changes F — and the slope the implicit differentiation calculator returns with it.

Check your own steps against the implicit differentiation calculator above.