A reliable method
Differentiate both sides with respect to x, collect terms containing dy/dx, then solve. Evaluate at a point that actually lies on the relation.
Worked foundation example
On x²+y²=13, find dy/dx at (2,3). Differentiate: 2x+2y(dy/dx)=0. Solve for dy/dx=−x/y. At the given point this is -2/3.
The common trap
Differentiating y² gives 2y dy/dx, not just 2y. Inverse slopes require a nonzero original derivative.
Connect the representations
At corresponding points the derivative of an inverse is the reciprocal of the original slope. The input values are exchanged.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare implicit and inverse reasoning
On x²+y²=25 at (3,4), differentiate the relation to obtain 2x+2yy′=0 and hence y′=−3/4. Treating y as a constant would remove the very dependence the problem asks about. In an inverse-function task, the roles of input and output exchange: for f(x)=x³, f(2)=8 and the derivative of its inverse at 8 is 1/f′(2)=1/12.
Transfer and explain
Transfer: on xy=12 at (3,4), the product rule gives y+xy′=0 and slope −4/3. Verify the point satisfies the relation before differentiating; a slope at an off-curve point is not justified.
On x²+y²=13, find dy/dx at (2,3).
- Differentiate: 2x+2y(dy/dx)=0.
- Solve for dy/dx=−x/y.
- At the given point this is -2/3.
Compare the other answer choices
Lose the minus sign when isolating dy/dx.
Reverse the roles of x and y.
Differentiate x² but ignore the coefficient on dy/dx.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus