Implicit Differentiation Calculator: Find dy/dx for Any Equation With Steps

Type an equation like x² + y² = 25 and get dy/dx, the tangent and normal line at a chosen point, and the second derivative y″, all with the chain-rule steps written out. The curve is graphed and the slope is re-checked numerically.

Implicit Differentiation Calculator

Pick a tab, type your numbers and press Calculate. Tap an example chip to see the working first.

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When You Need Implicit Differentiation

Some curves cannot be written as y = f(x). A circle x² + y² = 25 gives two values of y for most x, and the folium x³ + y³ = 6xy cannot be solved for y in any tidy way. Implicit differentiation avoids the problem: differentiate both sides as they are, treating y as an unknown function of x.

The one rule to remember is the chain rule. Whenever you differentiate a term with y, you get an extra y′ (dy/dx): d/dx[y²] = 2y·y′.

The Five-Step Method

  1. Differentiate both sides with respect to x.
  2. Attach y′ to every derivative of a y-term (chain rule).
  3. Collect all y′ terms on one side.
  4. Factor out y′.
  5. Divide to isolate it.
The circleTry it in the calculator ↑

x² + y² = 25, find dy/dx

  1. Differentiate: 2x + 2y·y′ = 0.
  2. Move: 2y·y′ = −2x.
  3. Divide: y′ = −x/y.

Answer: dy/dx = −x/y

Tangent line at (3, 4)Try it in the calculator ↑

Tangent to x² + y² = 25 at (3, 4)

  1. Check the point: 9 + 16 = 25 ✓.
  2. Slope: −x/y = −3/4.
  3. Point-slope: y − 4 = −(3/4)(x − 3), which simplifies to y = −(3/4)x + 25/4.

Answer: Tangent: y = −0.75x + 6.25. Normal slope: 4/3

A Harder Example: the Folium of Descartes

x³ + y³ = 6xyTry it in the calculator ↑

Find dy/dx and the slope at (3, 3)

  1. Differentiate: 3x² + 3y²·y′ = 6y + 6x·y′ (product rule on 6xy).
  2. Collect y′: 3y²·y′ − 6x·y′ = 6y − 3x².
  3. Factor and divide: y′ = (2y − x²) ÷ (y² − 2x).
  4. At (3, 3): (6 − 9) ÷ (9 − 6) = −1. Check the point: 27 + 27 = 54 = 6·9 ✓.

Answer: dy/dx = (2y − x²)/(y² − 2x), slope −1 at (3, 3)

Descartes proposed this curve in 1638 to test Fermat’s method of tangents. Three hundred years on it remains a good stress test because the answer depends on both x and y.

Second Derivative Implicitly

Differentiate y′ again, then substitute the first result and, if possible, the original equation to simplify.

Circle, second derivativeTry it in the calculator ↑

x² + y² = 25, find y″

  1. y′ = −x/y. Differentiate with the quotient rule: y″ = −(y − x·y′) ÷ y².
  2. Substitute y′ = −x/y: y″ = −(y + x²/y) ÷ y² = −(y² + x²) ÷ y³.
  3. Use x² + y² = 25.

Answer: y″ = −25/y³

The sign tells you something real: at the top of the circle (y > 0) y″ is negative, so the curve is concave down, and at the bottom it is concave up.

What We Think Is Overrated, Underrated and Wrong

Overrated: solving for y first when it is possible. It often creates square roots and ± signs that make the derivative messier than the implicit route.
Underrated: plugging the point in before simplifying. If you only need a slope, substitute x and y into the unsimplified equation and solve for y′ as a number.
Wrong: “dy/dx must be a function of x only”. For implicit curves it usually depends on both x and y.

Worked Examples With Real Numbers

For the circle x² + y² = 25, differentiate both sides: 2x + 2y·y′ = 0, so dy/dx = −x ÷ y. At (3, 4) the slope is −3/4, and the tangent line is y − 4 = −(3/4)(x − 3), or 3x + 4y = 25.

For the folium x³ + y³ = 6xy: 3x² + 3y²y′ = 6y + 6xy′, so dy/dx = (2y − x²) ÷ (y² − 2x). The tangent is vertical where y² = 2x.

Our Take: What Is Overrated, Underrated and Plain Wrong

Overrated: solving for y first. Many curves cannot be solved for y, and implicit differentiation does not need to.

Underrated: a slope check at a known point. On a circle, the slope must be perpendicular to the radius.

Plain wrong: forgetting the y′ factor. d/dx y² is 2y·y′, not 2y.

Implicit Differentiation Mistakes That Cost Marks

  • Forgetting y′ on a y-term: d/dx[y³] is 3y²·y′, not 3y².
  • Missing the product rule on mixed terms like xy.
  • Not checking that the point lies on the curve.
  • Dividing by zero: a vertical tangent occurs where the denominator is 0 (y = 0 on the circle).
  • Mixing up the tangent slope and the normal slope (negative reciprocal).

Why Students Get Stuck on Implicit Differentiation, and What a Good Calculator Should Do

Searching for help with implicit differentiation usually starts late at night with one problem that will not work out. Y is a function of x, but it is hidden inside the equation. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.

In short: students need to differentiate both sides and solve for dy/dx, not just see a final number, and a calculator earns trust only when its steps can be checked.

Understand the frustrations and desires

What they feel, and what they want

Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.

  • Wants each y term flagged with dy/dx
  • Wants the final dy/dx simplified
  • Wants the slope at a given point

Build the page around their intent

Each intent has a place on this page

The page follows intent, not features, so each need has a clear starting point.

  • I need an answer now: type the problem into the calculator at the top of this page.Find dy/dx for x^2 + y^2 = 25
  • I need to understand: the sections below explain the idea in plain words and list the common mistakes.
  • I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
  • I need practice: open the solver, tap + and choose Create Practice Test.
  • I need it explained aloud: in the solver, tap + and choose Create Video.

Implicit Differentiation: Questions People Ask

What is implicit differentiation?

Differentiating an equation in x and y directly, with y treated as a function of x, then solving for dy/dx.

Why do we multiply by dy/dx?

Because y depends on x, so the chain rule applies to every y-term.

How do I find a tangent line implicitly?

Compute dy/dx, evaluate it at the point, then use point-slope form.

What if the point is not on the curve?

The tangent is undefined. Substitute it into the equation first to check.

Can I find the second derivative?

Yes, switch to the second derivative mode. It reuses the original equation to simplify.

What does a vertical tangent look like?

The denominator of dy/dx is zero while the numerator is not.

How do I do implicit differentiation?

Differentiate both sides with respect to x, attach y′ to every derivative of a y-term, then solve for y′.

Why do I multiply by dy/dx?

y depends on x, so the chain rule gives d/dx f(y) = f′(y)·dy/dx.

How do I find the tangent line with implicit differentiation?

Compute dy/dx at the point and use y − y₀ = m(x − x₀).

What if the denominator of dy/dx is zero?

The tangent is vertical there, as at the left and right ends of a circle.

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