An implicit differentiation calculator finds dy/dx for equations where y is tangled up with x and cannot — or should not — be isolated on one side. Circles, ellipses, and curves like x³ + y³ = 6xy are not written as y = something, yet they still have slopes at (almost) every point, and this technique is how you get them.
Quick Answer
Differentiate both sides term by term, attaching dy/dx every time you differentiate a y-term, then solve the resulting equation for dy/dx. For the circle x² + y² = 25 this gives 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y — at the point (3, 4) the slope is −3/4.
Why y Carries an Extra Factor
When an equation is given implicitly, y is still secretly a function of x — the curve would not pass the "one y for nearby x" test everywhere, but locally it behaves like one. So differentiating y² with respect to x is really differentiating (y(x))², a composite function, and the chain rule demands the inner derivative: d/dx[y²] = 2y × dy/dx. That attached factor is the entire secret of the method. Terms in x alone differentiate as usual, constants vanish, and terms mixing x and y — like xy — need the product rule first, producing one piece with dy/dx and one without. Afterwards it is plain algebra: gather every term containing dy/dx on one side, factor it out, and divide.
The payoff is that you never had to solve for y. Solving x² + y² = 25 explicitly means y = ±√(25 − x²), two functions and a square root to differentiate; the implicit route is three lines. For curves like the folium x³ + y³ = 6xy, explicit solving is genuinely impractical, and implicit differentiation is not a shortcut but the only workable road. A bonus: the resulting dy/dx is expressed in both x and y, which is often simpler and is exactly what you need to evaluate the slope at a known point on the curve.
How to Use the Implicit Differentiation Calculator
- Enter the equation exactly as given, keeping both sides as they are — do not rearrange it into y = form first.
- Make sure the variable you want to differentiate with respect to (usually x) is set correctly.
- Read the resulting expression for dy/dx, which will typically contain both x and y.
- To find the slope at a specific point, substitute that point's coordinates into the expression.
- Check that your point actually lies on the curve before trusting the slope — substituting a point that is not on it gives a meaningless number.
- Compare each line with your own attempt, watching for where the dy/dx factors appear.
Worked Example: Slope of a Circle at (3, 4)
The circle x² + y² = 25 has radius 5, and (3, 4) lies on it because 9 + 16 = 25. Differentiating both sides with respect to x: 2x + 2y·(dy/dx) = 0. Solving, dy/dx = −x/y, and at (3, 4) the slope is −3/4. The tangent line there is y − 4 = −(3/4)(x − 3), or y = −(3/4)x + 6.25. There is a satisfying geometric check: the radius to (3, 4) has slope 4/3, and 4/3 × (−3/4) = −1, confirming the tangent is perpendicular to the radius, exactly as circle geometry demands. A slightly harder pattern works the same way: for x² + xy + y² = 7, differentiation gives 2x + y + x·(dy/dx) + 2y·(dy/dx) = 0, so dy/dx = −(2x + y) ÷ (x + 2y).
That mixed-term pattern deserves one complete run-through, because it is the form examiners actually set. Take the curve x² + xy + y² = 7 and the point (1, 2) — which lies on it, since 1 + 2 + 4 = 7. Differentiate everything with respect to x: the x² term gives 2x; the xy term needs the product rule and gives y + x·(dy/dx); the y² term gives 2y·(dy/dx); the constant gives 0. Collecting the dy/dx terms: dy/dx = −(2x + y) ÷ (x + 2y), as before. Now substitute the point: −(2 + 2) ÷ (1 + 4) = −4/5. The tangent at (1, 2) is therefore y − 2 = −(4/5)(x − 1). Two features to absorb: the final derivative still contains both variables, which is normal and necessary for implicit curves, and the product-rule term is what put the plain y into the numerator — leave it out and the slope comes out wrong by exactly that contribution.
The circle also hands you its horizontal tangents for free, which is a good test of whether the formula has been understood rather than memorised. The slope −x/y is zero exactly where the numerator is zero — at x = 0, giving the points (0, 5) and (0, −5), the very top and bottom of the circle, where a horizontal tangent is precisely what a sketch predicts. A formula that agrees with the picture at the extremes can be trusted in the middle; building this habit of testing a general result against a case you can see is worth more than any single answer the calculator returns.
Slip-Ups Specific to Implicit Work
The signature error is differentiating a y-term as if y were x: writing d/dx[y²] = 2y without the dy/dx factor destroys everything downstream. The second is forgetting the product rule on mixed terms like xy or x²y, which each produce two pieces. Third, sign errors while gathering terms — move the non-dy/dx terms across the equals sign carefully, one at a time. Fourth, points where the denominator of your final expression is zero: on the circle, dy/dx = −x/y blows up at (5, 0) and (−5, 0), which is not a failure of the method but information — the tangent there is vertical. Fifth, reporting a slope for a point that does not satisfy the original equation; always verify membership first, as the example did with 9 + 16 = 25.
The Chain Rule in Disguise
Every implicit differentiation problem is secretly an application of the Chain Rule Calculator's pattern — the dy/dx factor is precisely the "derivative of the inside" — and the results feed the same slope questions as the ordinary Derivative Calculator, just for curves that refuse to be written as functions. If you are working through a calculus course, practise the same problem both ways whenever an explicit form exists; agreement between the two answers is the fastest confidence check there is. More tools in this family are listed under Math Calculators.