Treating y as a function of x without ever solving for it, and what happens when two quantities change together.
x² + y² = 25 is a circle. It fails the vertical line test, so there is no single y = f(x) to differentiate. Move the point and the gradient still exists everywhere, and implicit differentiation finds it.
The tangent is always perpendicular to the radius, which is exactly what dy/dx = −x/y is saying.
Differentiate both sides with respect to x, treating y as a function of x. Every time you differentiate something containing y, the chain rule produces a dydx. Then collect those terms and make it the subject.
The answer contains y, and that is fine. An implicit derivative usually depends on both coordinates, which is why you need a point rather than just an x value. Students who expect a formula in x alone tend to assume they have gone wrong.
A term like xy needs the product rule as well:
A ladder of length 5 m leans against a wall. Its foot slides out at 0.5 m per second. How fast is the top sliding down when the foot is 3 m out?
Differentiate with respect to time, then substitute. Putting x = 3 in before differentiating turns a variable into a constant, and its rate of change vanishes. That is the commonest error on related rates, and the answer comes out as zero or nonsense.
1. For x² + y² = 25, find dydx at the point (4, 3).
2. Differentiating y³ with respect to x gives:
3. In the ladder problem, find dydt when the foot is 3 m out.
Producing a dydx every time you differentiate a y term. Missing one is the error that defines this sub-topic.
On related rates, writing the chain of derivatives before substituting any numbers, and differentiating with respect to t.
Interpreting the sign. A negative rate means decreasing, and saying so in words is often the final mark.
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