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Implicit Differentiation at a Point
If
x2+xy+y2=12
then at the point (2,2), dydx is
Need a hint?
Use product rule on xy.
Correct answer
B. −1
Solution
Differentiating implicitly gives 2x+xdydx+y+2ydydx=0. Grouping derivative terms gives (x+2y)dydx=−(2x+y). Thus dydx=−2x+yx+2y. At (2,2), this is −66=−1.
Detailed solution
The term xy requires product rule because both variables depend on x implicitly.
Common mistake
A common mistake is differentiating xy as only xdydx, forgetting the +y term.
Trap logic
The trap is missing product rule on xy.
What this reveals
If this was difficult, it may reveal weakness in implicit differentiation.
StudyGlitch notes
This AP Calculus AB differentiation question tests implicit differentiation.
For implicit differentiation, treat y as a function of x.
After differentiating, plug in (2,2).
This week’s question is more than a score.
A wrong answer usually points to a pattern: slow recognition, a trap choice, weak setup, or pressure under time. Use this challenge as a signal, then connect it to your diagnostic, practice history, and weekly StudyGlitch progress.