Key takeaways
- Differentiate both sides as they are, without solving for y first.
- Every time you differentiate a y term, multiply by dy/dx.
- For a circle x² + y² = r², the slope is dy/dx = −x/y.
- The #1 mistake is forgetting dy/dx on the y terms.
In this guide
What is implicit differentiation?
Usually a curve is written with y alone on one side, like y = x². That is called explicit. Some curves are easier to write with x and y mixed together, like a circle: x² + y² = 25. That is called implicit.
You still want the slope, dy/dx, at a point. Instead of solving for y first, you differentiate both sides as they are, remembering that y changes when x changes.
Why does implicit differentiation matter?
Solving for y can be messy or impossible. For a circle, y = ±√(r² − x²) splits into two halves. For curves like x³ + y³ = 6xy, there is no neat “y = …” formula at all. Implicit differentiation works on the equation just as it is.
The radius of a circle and its tangent always meet at a right angle. Implicit differentiation proves it in one line: the radius has slope y/x, the tangent has slope −x/y, and those multiply to −1.
The formula
For a circle of radius r centred at the origin:
| Symbol | What it means | Example |
|---|---|---|
| r | The radius of the circle | 13 |
| x₀ | The x-coordinate of your point on the circle | 5 |
| y₀ | The y-coordinate, found from √(r² − x₀²) | √(169 − 25) = 12 |
| dy/dx | The slope of the tangent line at that point | −5/12 ≈ −0.4167 |
This calculator takes r and x₀, finds y₀ on the top half of the circle (where y is positive), and returns −x₀/y₀. You can check y₀ with the Pythagorean theorem calculator.
How to do implicit differentiation step by step
- 1Differentiate every term with respect to x. x² becomes 2x.
- 2Multiply y terms by dy/dx. y² becomes
2y·dy/dx. This is the chain rule at work. - 3Constants become 0. r² is a fixed number, so you get
2x + 2y·dy/dx = 0. - 4Solve for dy/dx.
dy/dx = −2x ÷ 2y = −x/y. - 5Put in your point. Use both x and y.
Now try it yourself. Enter the radius and an x-value in the calculator below to get the slope on the top half of the circle.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using the wrong half of the circle. At (3, −4), the slope is
−3/(−4) = +0.75. The calculator uses the top half, where y is positive. - Choosing x = ±r. At the far left and right of the circle, y = 0 and the tangent is vertical, so the slope is undefined. The calculator says so.
- Picking x bigger than r. Then the point is not on the circle at all, and the calculator shows an error.
Tips and tricks
- Other curves: for
xy = 6, the product rule givesy + x·dy/dx = 0, sody/dx = −y/x. At (2, 3) the slope is −1.5. - Top of the circle: at x = 0 the slope is 0, a flat tangent. That’s a quick sanity check.
- Straight lines: for slopes between two points, the slope formula is all you need.
Where you’ll use it in real life
- Related rates: how fast the top of a ladder slides down a wall as the bottom slides out.
- Engineering: slopes on circular and elliptical parts, like tracks, arches and gears.
- Economics: curves where two quantities are linked, such as equal-satisfaction curves.
- Proofs: deriving rules like the derivative of ln x from eʸ = x.
Quick summary
Implicit differentiation finds dy/dx without solving for y: differentiate every term, multiply each y term by dy/dx, then solve. For a circle x² + y² = r², the slope is −x/y. Use the calculator above for any point on the top half of a circle, with every step shown.
Worked example
A circular running track has radius 13 m, with its centre at the origin. A runner is at the point on the top half of the track where x = 5 m. What is the slope of the track (the runner’s direction) at that spot? Here r = 13 and x₀ = 5.
- 01Circle equation: x² + y² = 13² = 169
- 02Differentiating implicitly: 2x + 2y(dy/dx) = 0
- 03Solve for dy/dx: dy/dx = −x/y
- 04At x₀ = 5: y₀ = √(13² − 5²) = √144 = 12
- 05dy/dx = −5 / 12 = -0.4167
These numbers are pre-filled in the calculator above.
Frequently asked questions
When should I use implicit differentiation?
Use it when x and y are mixed together and it is hard or impossible to get y on its own, such as x² + y² = 25 or x³ + y³ = 6xy.
Why do I multiply by dy/dx?
Because y is itself a function of x. By the chain rule, differentiating anything in y gives its normal derivative times dy/dx.
Why is the slope of a circle −x/y?
The radius from the centre to (x, y) has slope y/x, and the tangent line is at right angles to the radius. Perpendicular slopes multiply to −1, so the tangent slope is −x/y.
