APFIVE Implicit Differentiation Error Analysis
A student differentiates the equation $$xy + y^2 = 10$$ with respect to x. She applies the product rule only to y^2 and obtains $$y + y' + 2*y*y' = 0$$, omitting the necessary derivative term for xy. Identify the mistake in her process.
TABLE: Differentiation of Products
| Expression | Correct Derivative |
|---|---|
| x*y | x*(dy/dx) + y |
| y^2 | 2y(dy/dx) |
A
There is no mistake; her differentiation is correct.
B
She should have treated xy + y^2 as a constant multiple of y.
C
The error is in differentiating y^2 rather than xy.
D
The student omitted the product rule for xy; the correct differentiation is x*(dy/dx) + y + 2y(dy/dx) = 0, so that dy/dx = -y/(x + 2*y).
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