Implicit Differentiation x²+y²=25 (Circle)

The equation x² + y² = 25 describes a circle of radius 5 centered at the origin. Students find implicit differentiation confusing for one reason: y is trapped inside the equation, and you cannot just solve for y and differentiate normally (well, you could, but you would get two messy square-root functions, one for the top half and one for the bottom half).

Implicit differentiation differentiates the equation exactly as it stands, treating y as a function of x and using the chain rule whenever y appears. The prize at the end is dy/dx = −x/y: a tiny formula that hands you the slope of the circle at any point you like.

We will show that differentiating x² + y² = 25 gives dy/dx = −x/y, and that at the point (3, 4) the slope is −3/4.

Worked solution

Step 1: Differentiate both sides with respect to x.

d/dx (x² + y²) = d/dx (25)

Take it term by term.

Step 2: Differentiate x².

d/dx (x²) = 2x. Nothing special here; it is the ordinary power rule.

Step 3: Differentiate y² using the chain rule.

This is the step that makes the method "implicit." y is secretly a function of x, so y² is a composition: first y depends on x, then you square it. The chain rule says: differentiate the outside, then multiply by the derivative of the inside.

Outside: d/dy (y²) = 2y
Inside: d/dx (y) = dy/dx

So d/dx (y²) = 2y · dy/dx. Every time you differentiate a y-term, a factor of dy/dx appears. That is the whole game.

Step 4: Differentiate the right side.

d/dx (25) = 0, because 25 is a constant.

Step 5: Assemble the differentiated equation.

2x + 2y · dy/dx = 0

Step 6: Solve for dy/dx.

2y · dy/dx = −2x
dy/dx = −2x / 2y = −x/y ✓

Step 7: Use the formula at a point.

Take the point (3, 4). First confirm it is on the circle: 3² + 4² = 9 + 16 = 25 ✓. Then substitute x = 3, y = 4:

dy/dx = −3/4

So the tangent to the circle at (3, 4) has slope −3/4. If you sketch it, the circle at that point is falling gently to the right, which matches a negative slope smaller than 1 in size.

Step 8: Check (never skip).

Two checks. First, the algebra: differentiate −x/y back? Better, verify with geometry. The radius from the origin to (3, 4) has slope 4/3. The tangent line must be perpendicular to the radius, and perpendicular slopes are negative reciprocals: the negative reciprocal of 4/3 is −3/4. That matches our dy/dx exactly ✓. Second, test another point: at (5, 0), the formula gives dy/dx = −5/0, which is undefined, meaning a vertical tangent. Look at the circle: at (5, 0), the rightmost point, the tangent is indeed vertical ✓. The formula behaves correctly even at the edge case.

Why this method works

Implicit differentiation is the chain rule applied honestly. Writing dy/dx every time you differentiate a y-term is just admitting that y depends on x. The beautiful payoff in this example is geometric: dy/dx = −x/y says the tangent slope is the negative reciprocal of the radius slope y/x, which is exactly the perpendicularity you expect from circle geometry. Calculus and geometry agree, which is why the answer −x/y feels inevitable rather than accidental.

Common mistakes

Forgetting the dy/dx factor on y-terms. Writing d/dx (y²) = 2y and stopping there is the classic error. The chain rule demands the extra dy/dx: d/dx (y²) = 2y · dy/dx. Say "dee why dee ex" out loud each time you differentiate a y-term until it is automatic.
Sign errors when isolating dy/dx. From 2x + 2y·dy/dx = 0, moving 2x across gives −2x, not +2x. Rushing this line flips the sign of the whole answer.
Plugging the point in before solving for dy/dx. Substitute (3, 4) only after you have the general formula dy/dx = −x/y. If you plug numbers in too early, you lose the formula and cannot answer follow-up questions about other points.

FAQ

Q: What does dy/dx = −x/y actually mean?
A: It is the slope of the tangent line to the circle at whatever point (x, y) you choose. At (3, 4) the slope is −3/4; at (4, 3) it would be −4/3. One formula answers infinitely many slope questions.

Q: Why not just solve for y first and differentiate normally?
A: You would get y = ±√(25 − x²), two functions, and differentiating gives ∓x/√(25 − x²). At (3, 4) that is −3/4, the same answer, but with much more work and an ugly formula. Implicit differentiation skips the solving step entirely.

Q: What happens when y = 0 in the formula?
A: You get division by zero, which means the slope is undefined: the tangent is vertical. On the circle x² + y² = 25 this happens at (5, 0) and (−5, 0), the leftmost and rightmost points, where the tangent is indeed vertical. The formula does not break; it tells you the truth.

Differentiate implicitly with steps on our implicit derivative calculator.